ISEE Upper Level Quiz: Factoring Expressions
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Factoring ExpressionsQuestion 1 of 20

Which of the following expressions is equivalent to 6x3+9x2−6x−96x^3 + 9x^2 - 6x - 9?

3(2x3+3x2−2x−3)3(2x^3 + 3x^2 - 2x - 3)
3(x+1)(2x2−3)3(x + 1)(2x^2 - 3)
3(x−1)(2x2+5x+3)3(x - 1)(2x^2 + 5x + 3)
3(2x+3)(x2−1)3(2x + 3)(x^2 - 1)
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ISEE Upper Level Quiz

ISEE Upper Level Quiz: Factoring Expressions

Practice Factoring Expressions in ISEE Upper Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Factoring Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which of the following expressions is equivalent to 6x3+9x2−6x−96x^3 + 9x^2 - 6x - 9?

  1. 3(2x3+3x2−2x−3)3(2x^3 + 3x^2 - 2x - 3)
  2. 3(x+1)(2x2−3)3(x + 1)(2x^2 - 3)
  3. 3(x−1)(2x2+5x+3)3(x - 1)(2x^2 + 5x + 3)
  4. 3(2x+3)(x2−1)3(2x + 3)(x^2 - 1) (correct answer)
Explanation: When you see a polynomial expression like this, you're being tested on factoring techniques. The goal is to break down the original expression into simpler factors that, when multiplied together, give you back the original polynomial. Let's start by looking for a common factor in 6x3+9x2−6x−96x^3 + 9x^2 - 6x - 9. Each term is divisible by 3, so we can factor out 3: 3(2x3+3x2−2x−3)3(2x^3 + 3x^2 - 2x - 3). Now we need to factor the expression inside the parentheses further. For 2x3+3x2−2x−32x^3 + 3x^2 - 2x - 3, we can use grouping. Group the first two terms and the last two terms: x2(2x+3)−1(2x+3)x^2(2x + 3) - 1(2x + 3). Notice that (2x+3)(2x + 3) is a common factor, so we get (2x+3)(x2−1)(2x + 3)(x^2 - 1). Since x2−1x^2 - 1 is a difference of squares, we could factor it further as (x+1)(x−1)(x + 1)(x - 1), but none of our answer choices do this. Therefore, our final answer is 3(2x+3)(x2−1)3(2x + 3)(x^2 - 1), which matches choice D. Choice A stops at the first step of factoring out 3. Choice B gives 3(x+1)(2x2−3)3(x + 1)(2x^2 - 3), which expands to 6x3+6x2−9x−96x^3 + 6x^2 - 9x - 9 (incorrect middle terms). Choice C gives 3(x−1)(2x2+5x+3)3(x - 1)(2x^2 + 5x + 3), which expands to 6x3+9x2−9x−96x^3 + 9x^2 - 9x - 9 (wrong sign on the xx term). Remember: always verify factoring by expanding your answer back to check if it matches the original expression.

Question 2

If 3x2+bx+123x^2 + bx + 12 factors as (3x+4)(x+3)(3x + 4)(x + 3), what is the value of bb?

  1. b=9b = 9
  2. b=11b = 11
  3. b=13b = 13 (correct answer)
  4. b=15b = 15
Explanation: When you see a factoring problem where you're given the factored form and need to find a coefficient, you should expand the factored expression and compare it to the original polynomial. Let's expand (3x+4)(x+3)(3x + 4)(x + 3) using FOIL or the distributive property: (3x+4)(x+3)=3x⋅x+3x⋅3+4⋅x+4⋅3(3x + 4)(x + 3) = 3x \cdot x + 3x \cdot 3 + 4 \cdot x + 4 \cdot 3 =3x2+9x+4x+12= 3x^2 + 9x + 4x + 12 =3x2+13x+12= 3x^2 + 13x + 12 Since this must equal 3x2+bx+123x^2 + bx + 12, we can see that b=13b = 13, making C the correct answer. Let's examine why the other choices are wrong. Choice A (b=9b = 9) might come from only considering the first cross-multiplication term (3x⋅3=9x3x \cdot 3 = 9x) and forgetting the second term. Choice B (b=11b = 11) doesn't correspond to any natural step in the expansion process—it's likely a distractor. Choice D (b=15b = 15) might result from incorrectly multiplying the constant terms of the binomials (4×3=124 \times 3 = 12, but somehow getting confused and using 4+3+4+3+1=154 + 3 + 4 + 3 + 1 = 15 or another computational error). Remember: when working with factored polynomials, always expand completely and collect like terms. The middle term in a quadratic comes from adding the two cross-multiplication terms when you FOIL. Double-check your arithmetic, especially when combining like terms—this is where most errors occur in factoring problems.

Question 3

What is the factored form of x2+7x+12x^2 + 7x + 12?

  1. (x+3)(x+4)(x + 3)(x + 4) (correct answer)
  2. (x−3)(x−4)(x - 3)(x - 4)
  3. (x+2)(x+6)(x + 2)(x + 6)
  4. (x+1)(x+12)(x + 1)(x + 12)
Explanation: When you see a quadratic expression like x2+7x+12x^2 + 7x + 12, you're looking at a factoring problem. The goal is to find two binomials that multiply together to give you the original expression. To factor x2+7x+12x^2 + 7x + 12, you need two numbers that multiply to give you the constant term (12) and add to give you the middle coefficient (7). Let's think systematically: what pairs of numbers multiply to 12? The possibilities are 1×12, 2×6, and 3×4. Now check which pair adds to 7: 1+12=13, 2+6=8, and 3+4=7. Perfect! The numbers 3 and 4 work. Since both numbers are positive (they add to positive 7 and multiply to positive 12), both signs in the factored form will be positive: (x+3)(x+4)(x + 3)(x + 4). You can verify this by expanding: (x+3)(x+4)=x2+4x+3x+12=x2+7x+12(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12. This confirms answer choice A is correct. Looking at the wrong answers: Choice B has (x−3)(x−4)(x - 3)(x - 4), which expands to x2−7x+12x^2 - 7x + 12 — the wrong sign on the middle term. Choice C gives (x+2)(x+6)=x2+8x+12(x + 2)(x + 6) = x^2 + 8x + 12 — wrong middle coefficient. Choice D gives (x+1)(x+12)=x2+13x+12(x + 1)(x + 12) = x^2 + 13x + 12 — also wrong middle coefficient. Remember: when factoring x2+bx+cx^2 + bx + c, find two numbers that multiply to cc and add to bb. The signs of these numbers determine the signs in your factors.

Question 4

If x2+mx+n=(x+5)(x−2)x^2 + mx + n = (x + 5)(x - 2), what is the value of m+nm + n?

  1. −13-13
  2. −7-7 (correct answer)
  3. 33
  4. 77
Explanation: This question tests your ability to expand binomial expressions and match coefficients. When you see an equation where a quadratic expression equals a factored form, you need to expand the right side and compare coefficients. Let's expand (x+5)(x−2)(x + 5)(x - 2) using FOIL or the distributive property: (x+5)(x−2)=x2−2x+5x−10=x2+3x−10(x + 5)(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10 Now we can match this to the left side x2+mx+nx^2 + mx + n:
  • The coefficient of x2x^2 is 1 (matches)
  • The coefficient of xx is m=3m = 3
  • The constant term is n=−10n = -10
Therefore, m+n=3+(−10)=−7m + n = 3 + (-10) = -7. Looking at the wrong answers: Choice A gives −13-13, which you might get if you incorrectly calculated m⋅n=3×(−10)=−30m \cdot n = 3 \times (-10) = -30 instead of m+nm + n, or made an arithmetic error. Choice C gives 33, which is just the value of mm alone—a common mistake when students forget to include nn. Choice D gives 77, which you'd get if you calculated m+nm + n but made a sign error, treating nn as positive 10 instead of negative 10. Strategy tip: When expanding factored expressions, always double-check your signs, especially with subtraction. Write out each step clearly: (x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a+b)x + ab. The middle coefficient is the sum of the constants, and the last term is their product.

Question 5

What is the complete factorization of 8x3−278x^3 - 27?

  1. (2x−3)(4x2+6x+9)(2x - 3)(4x^2 + 6x + 9) (correct answer)
  2. (2x+3)(4x2−6x+9)(2x + 3)(4x^2 - 6x + 9)
  3. (2x−3)(4x2−6x+9)(2x - 3)(4x^2 - 6x + 9)
  4. (2x−3)3(2x - 3)^3
Explanation: When you see an expression like 8x3−278x^3 - 27, you should recognize this as a difference of cubes pattern: a3−b3a^3 - b^3. The key insight is identifying what's being cubed: 8x3=(2x)38x^3 = (2x)^3 and 27=3327 = 3^3. The difference of cubes formula is a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2). With a=2xa = 2x and b=3b = 3, we get: 8x3−27=(2x)3−33=(2x−3)((2x)2+(2x)(3)+32)8x^3 - 27 = (2x)^3 - 3^3 = (2x - 3)((2x)^2 + (2x)(3) + 3^2) Simplifying the second factor: (2x)2+(2x)(3)+32=4x2+6x+9(2x)^2 + (2x)(3) + 3^2 = 4x^2 + 6x + 9 Therefore: 8x3−27=(2x−3)(4x2+6x+9)8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9) Choice A matches this exactly and is correct. Choice B has (2x+3)(2x + 3) as the first factor, which would be wrong because we have subtraction, not addition, in our original expression. This represents the sum of cubes pattern instead. Choice C has the correct first factor but shows 4x2−6x+94x^2 - 6x + 9 in the second factor. The middle term should be +6x+6x, not −6x-6x, based on the difference of cubes formula. Choice D shows (2x−3)3(2x - 3)^3, which would equal 8x3−36x2+54x−278x^3 - 36x^2 + 54x - 27—a completely different expression with additional terms. Study tip: Memorize both factoring formulas: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2) and a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2). Notice how the signs alternate in specific patterns.

Question 6

If ax2+bx+c=(3x−2)(2x+1)ax^2 + bx + c = (3x - 2)(2x + 1), what is the value of a−b+ca - b + c?

  1. 33
  2. 55 (correct answer)
  3. 77
  4. 99
Explanation: When you see an equation where a quadratic expression equals a factored form, you're being asked to expand the right side and match coefficients. This tests your ability to multiply binomials and understand how the coefficients aa, bb, and cc relate to the factored form. To find aa, bb, and cc, expand (3x−2)(2x+1)(3x - 2)(2x + 1) using FOIL:
  • First: 3x⋅2x=6x23x \cdot 2x = 6x^2
  • Outer: 3x⋅1=3x3x \cdot 1 = 3x
  • Inner: (−2)⋅2x=−4x(-2) \cdot 2x = -4x
  • Last: (−2)⋅1=−2(-2) \cdot 1 = -2
Combining like terms: 6x2+3x−4x−2=6x2−x−26x^2 + 3x - 4x - 2 = 6x^2 - x - 2 Therefore, a=6a = 6, b=−1b = -1, and c=−2c = -2. Now calculate a−b+c=6−(−1)+(−2)=6+1−2=5a - b + c = 6 - (-1) + (-2) = 6 + 1 - 2 = 5. Choice A (33) might result from incorrectly identifying a=3a = 3 by looking only at the first coefficient in the factored form. Choice C (77) could come from sign errors when combining terms or miscalculating 6−(−1)−(−2)6 - (-1) - (-2). Choice D (99) might occur if you mistakenly calculated a+b+ca + b + c instead of a−b+ca - b + c, getting 6+(−1)+(−2)=36 + (-1) + (-2) = 3, though even that doesn't equal 99. The correct answer is B. Study tip: When expanding factored quadratics, always double-check your signs, especially with subtraction. Also, remember that a−b+ca - b + c is actually the value of the quadratic when x=−1x = -1, which can serve as a quick verification method.

Question 7

What is the greatest common factor of the terms in 12x3y2+18x2y3−6xy412x^3y^2 + 18x^2y^3 - 6xy^4?

  1. 3xy3xy
  2. 6xy6xy
  3. 6xy26xy^2 (correct answer)
  4. 6x2y26x^2y^2
Explanation: When you encounter a problem asking for the greatest common factor (GCF) of polynomial terms, you need to find the largest expression that divides evenly into all terms. This means examining both the numerical coefficients and the variable parts separately. Let's break down each term in 12x3y2+18x2y3−6xy412x^3y^2 + 18x^2y^3 - 6xy^4:
  • First term: 12x3y212x^3y^2
  • Second term: 18x2y318x^2y^3
  • Third term: 6xy46xy^4
For the coefficients (12, 18, -6), find their GCF by listing factors. The GCF of 12, 18, and 6 is 6. For the variables, take the lowest power of each variable that appears in all terms:
  • For xx: the powers are 3, 2, and 1, so take x1=xx^1 = x
  • For yy: the powers are 2, 3, and 4, so take y2y^2
Therefore, the GCF is 6xy26xy^2, which is answer C. Looking at the wrong answers: A) 3xy3xy uses 3 instead of 6 for the coefficient—this fails to find the greatest numerical factor. B) 6xy6xy correctly identifies the coefficient as 6 but uses y1y^1 instead of y2y^2—remember to take the lowest power that appears in all terms. D) 6x2y26x^2y^2 incorrectly uses x2x^2 when the third term only has x1x^1, so x2x^2 doesn't divide evenly into all terms. Strategy tip: Always check your GCF by dividing it into each original term—if any division leaves a remainder or fraction, your GCF is too large.

Question 8

If 2x2−7x−4=(2x+1)(x+k)2x^2 - 7x - 4 = (2x + 1)(x + k) for some constant kk, what is the value of kk?

  1. k=−4k = -4 (correct answer)
  2. k=−2k = -2
  3. k=2k = 2
  4. k=4k = 4
Explanation: When you see an equation where a quadratic expression equals a factored form, you're working with polynomial factorization. The key is to expand the right side and match coefficients with the left side. Let's expand (2x+1)(x+k)(2x + 1)(x + k) using the distributive property: (2x+1)(x+k)=2x⋅x+2x⋅k+1⋅x+1⋅k=2x2+2kx+x+k=2x2+(2k+1)x+k(2x + 1)(x + k) = 2x \cdot x + 2x \cdot k + 1 \cdot x + 1 \cdot k = 2x^2 + 2kx + x + k = 2x^2 + (2k + 1)x + k Now we can match this with the original expression 2x2−7x−42x^2 - 7x - 4:
  • The x2x^2 coefficients match: 2=22 = 2 ✓
  • The xx coefficients must match: 2k+1=−72k + 1 = -7
  • The constant terms must match: k=−4k = -4
From the xx coefficient equation: 2k+1=−72k + 1 = -7, so 2k=−82k = -8, which gives us k=−4k = -4. We can verify this with the constant term: k=−4k = -4 matches perfectly. Let's check why the other answers fail. Choice B gives k=−2k = -2, which would make the xx coefficient 2(−2)+1=−32(-2) + 1 = -3, not −7-7. Choice C gives k=2k = 2, making the xx coefficient 2(2)+1=52(2) + 1 = 5, not −7-7. Choice D gives k=4k = 4, making the xx coefficient 2(4)+1=92(4) + 1 = 9, not −7-7. Therefore, A is correct. Study tip: When factoring quadratics, always expand your answer to verify it matches the original expression. This double-check catches arithmetic errors and confirms your factorization is correct.

Question 9

Which expression is equivalent to 9x2−30x+259x^2 - 30x + 25?

  1. (3x−5)2(3x - 5)^2 (correct answer)
  2. (9x−5)2(9x - 5)^2
  3. (3x−25)2(3x - 25)^2
  4. (3x+5)2(3x + 5)^2
Explanation: When you see a quadratic expression like 9x2−30x+259x^2 - 30x + 25, you should immediately think about factoring, particularly checking if it's a perfect square trinomial. Perfect square trinomials follow the pattern a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2 or a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2. To verify if this is a perfect square trinomial, identify what could be squared to give you the first and last terms. Since 9x2=(3x)29x^2 = (3x)^2 and 25=5225 = 5^2, we have potential factors of 3x3x and 55. Now check if the middle term fits: 2⋅3x⋅5=30x2 \cdot 3x \cdot 5 = 30x. Since our expression has −30x-30x, this confirms we have (3x−5)2(3x - 5)^2. You can verify by expanding: (3x−5)2=(3x)2−2(3x)(5)+52=9x2−30x+25(3x - 5)^2 = (3x)^2 - 2(3x)(5) + 5^2 = 9x^2 - 30x + 25 ✓ Choice A is correct. Choice B, (9x−5)2(9x - 5)^2, would expand to 81x2−90x+2581x^2 - 90x + 25, giving you the wrong coefficient for x2x^2. Choice C, (3x−25)2(3x - 25)^2, would expand to 9x2−150x+6259x^2 - 150x + 625, producing incorrect coefficients for both the middle and constant terms. Choice D, (3x+5)2(3x + 5)^2, would expand to 9x2+30x+259x^2 + 30x + 25, giving you a positive middle term instead of negative. Strategy tip: When factoring quadratics, always look for perfect square trinomials first by checking if the first and last terms are perfect squares, then verify the middle term equals twice their product.

Question 10

What is the factored form of x3+27x^3 + 27?

  1. (x+3)(x2+9)(x + 3)(x^2 + 9)
  2. (x+3)(x2−3x+9)(x + 3)(x^2 - 3x + 9) (correct answer)
  3. (x−3)(x2+3x+9)(x - 3)(x^2 + 3x + 9)
  4. (x+3)3(x + 3)^3
Explanation: When you encounter an expression like x3+27x^3 + 27, you're looking at a sum of cubes, which follows the special factoring pattern: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2). Recognizing this pattern is crucial because sum of cubes cannot be factored using simpler methods. First, identify the cube roots: x3=(x)3x^3 = (x)^3 and 27=3327 = 3^3, so we have a=xa = x and b=3b = 3. Applying the sum of cubes formula: x3+27=(x+3)(x2−x⋅3+32)=(x+3)(x2−3x+9)x^3 + 27 = (x + 3)(x^2 - x \cdot 3 + 3^2) = (x + 3)(x^2 - 3x + 9). You can verify this by expanding: (x+3)(x2−3x+9)=x3−3x2+9x+3x2−9x+27=x3+27(x + 3)(x^2 - 3x + 9) = x^3 - 3x^2 + 9x + 3x^2 - 9x + 27 = x^3 + 27 ✓ Choice A (x+3)(x2+9)(x + 3)(x^2 + 9) incorrectly omits the middle term −3x-3x from the quadratic factor. This is a common error when students forget the complete sum of cubes pattern. Choice C (x−3)(x2+3x+9)(x - 3)(x^2 + 3x + 9) uses the difference of cubes pattern instead. This would factor x3−27x^3 - 27, not x3+27x^3 + 27. Choice D (x+3)3(x + 3)^3 represents a perfect cube, which would expand to x3+9x2+27x+27x^3 + 9x^2 + 27x + 27—completely different from our original expression. Study tip: Memorize both cube factoring formulas: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2). Notice the sign patterns carefully—they're easy to mix up under test pressure.

Question 11

If 6x2+kx−5=(2x−1)(3x+5)6x^2 + kx - 5 = (2x - 1)(3x + 5), what is the value of kk?

  1. k=4k = 4
  2. k=7k = 7 (correct answer)
  3. k=10k = 10
  4. k=13k = 13
Explanation: When you see an equation where a quadratic expression equals a factored form, you're looking at a polynomial identity problem. The key insight is that both sides must be equivalent for all values of the variable, so you can expand the right side and compare coefficients. Let's expand the right side: (2x−1)(3x+5)(2x - 1)(3x + 5). Using FOIL: First terms give 2x⋅3x=6x22x \cdot 3x = 6x^2, Outer terms give 2x⋅5=10x2x \cdot 5 = 10x, Inner terms give −1⋅3x=−3x-1 \cdot 3x = -3x, and Last terms give −1⋅5=−5-1 \cdot 5 = -5. Combining like terms: 6x2+10x−3x−5=6x2+7x−56x^2 + 10x - 3x - 5 = 6x^2 + 7x - 5. Now we can match coefficients with the left side 6x2+kx−56x^2 + kx - 5. The x2x^2 coefficients match (both are 6), the constant terms match (both are -5), and the xx coefficients must match, so k=7k = 7. Looking at the wrong answers: A) k=4k = 4 would give you 6x2+4x−56x^2 + 4x - 5, which doesn't match our expansion. C) k=10k = 10 might tempt you if you only looked at the "outer" term (10x10x) but forgot to subtract the "inner" term (−3x-3x). D) k=13k = 13 could result from incorrectly adding 10+3=1310 + 3 = 13 instead of 10−3=710 - 3 = 7. The correct answer is B) k=7k = 7. Strategy tip: When checking polynomial identities, always expand completely and combine like terms before comparing coefficients. Don't try to shortcut by looking at individual terms from FOIL.

Question 12

If ax2+bx+cax^2 + bx + c factors as (4x−3)(2x+1)(4x - 3)(2x + 1), what is the value of abcabc?

  1. −48-48
  2. −24-24
  3. 2424
  4. 4848 (correct answer)
Explanation: This question tests your ability to work backward from a factored form to find the coefficients of a quadratic expression, then use those coefficients to calculate a specific value. To find the values of aa, bb, and cc, you need to expand the given factorization (4x−3)(2x+1)(4x - 3)(2x + 1). Using FOIL:
  • First: 4x⋅2x=8x24x \cdot 2x = 8x^2
  • Outer: 4x⋅1=4x4x \cdot 1 = 4x
  • Inner: −3⋅2x=−6x-3 \cdot 2x = -6x
  • Last: −3⋅1=−3-3 \cdot 1 = -3
Combining like terms: 8x2+4x−6x−3=8x2−2x−38x^2 + 4x - 6x - 3 = 8x^2 - 2x - 3 Therefore, a=8a = 8, b=−2b = -2, and c=−3c = -3, giving us abc=8⋅(−2)⋅(−3)=48abc = 8 \cdot (-2) \cdot (-3) = 48. The answer is D. Let's examine why the other choices are incorrect. Choice A (−48-48) likely comes from making a sign error during the multiplication—perhaps calculating 8⋅2⋅38 \cdot 2 \cdot 3 but incorrectly applying a negative sign. Choice B (−24-24) might result from correctly finding a=8a = 8 and c=−3c = -3 but miscalculating bb as −1-1 instead of −2-2, giving 8⋅(−1)⋅(−3)=248 \cdot (-1) \cdot (-3) = 24, then incorrectly applying a negative sign. Choice C (2424) could come from the same coefficient error as B but without the extra negative sign. When expanding factored quadratics, always double-check your arithmetic, especially with negative signs. The FOIL method is reliable, but carefully combine like terms and watch for sign errors when multiplying negative coefficients.

Question 13

Which expression is equivalent to 16x4−116x^4 - 1?

  1. (4x2+1)(4x2−1)(4x^2 + 1)(4x^2 - 1)
  2. (4x+1)(4x−1)(4x + 1)(4x - 1)
  3. (4x2+1)(2x+1)(2x−1)(4x^2 + 1)(2x + 1)(2x - 1) (correct answer)
  4. (2x+1)2(2x−1)2(2x + 1)^2(2x - 1)^2
Explanation: When you see an expression like 16x4−116x^4 - 1, recognize this as a difference of squares pattern: a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b). Here, 16x4=(4x2)216x^4 = (4x^2)^2 and 1=121 = 1^2, so you can factor this as (4x2+1)(4x2−1)(4x^2 + 1)(4x^2 - 1). However, you're not done yet! Notice that 4x2−14x^2 - 1 is itself another difference of squares: 4x2=(2x)24x^2 = (2x)^2 and 1=121 = 1^2. This means 4x2−1=(2x+1)(2x−1)4x^2 - 1 = (2x + 1)(2x - 1). So the complete factorization is: 16x4−1=(4x2+1)(4x2−1)=(4x2+1)(2x+1)(2x−1)16x^4 - 1 = (4x^2 + 1)(4x^2 - 1) = (4x^2 + 1)(2x + 1)(2x - 1), which is choice C. Choice A stops at the first factorization and doesn't factor 4x2−14x^2 - 1 further. While 16x4−1=(4x2+1)(4x2−1)16x^4 - 1 = (4x^2 + 1)(4x^2 - 1) is correct, it's not completely factored. Choice B incorrectly treats 16x416x^4 as (4x)2(4x)^2 instead of (4x2)2(4x^2)^2. If you expand (4x+1)(4x−1)(4x + 1)(4x - 1), you get 16x2−116x^2 - 1, not 16x4−116x^4 - 1. Choice D has the wrong exponents. Expanding (2x+1)2(2x−1)2(2x + 1)^2(2x - 1)^2 gives you [(2x+1)(2x−1)]2=(4x2−1)2[(2x + 1)(2x - 1)]^2 = (4x^2 - 1)^2, which equals 16x4−8x2+116x^4 - 8x^2 + 1, not our original expression. Always factor completely when possible. With difference of squares, check if either factor can be factored further using the same pattern.

Question 14

If x2+bx+20x^2 + bx + 20 factors as (x+4)(x+5)(x + 4)(x + 5), what is the value of bb?

  1. b=1b = 1
  2. b=9b = 9 (correct answer)
  3. b=10b = 10
  4. b=20b = 20
Explanation: When you see a factored quadratic expression, you can use the relationship between the factors and the original polynomial to find missing coefficients. The key is understanding how FOIL (First, Outer, Inner, Last) connects the factored form back to standard form. Let's expand (x+4)(x+5)(x + 4)(x + 5) using FOIL:
  • First: x⋅x=x2x \cdot x = x^2
  • Outer: x⋅5=5xx \cdot 5 = 5x
  • Inner: 4⋅x=4x4 \cdot x = 4x
  • Last: 4⋅5=204 \cdot 5 = 20
Combining these terms: x2+5x+4x+20=x2+9x+20x^2 + 5x + 4x + 20 = x^2 + 9x + 20 Since this must equal x2+bx+20x^2 + bx + 20, we can see that b=9b = 9. Looking at the wrong answers: Choice (A) gives b=1b = 1, which would mean the middle term is just xx—far too small compared to our sum of 4x+5x4x + 5x. Choice (C) suggests b=10b = 10, which might tempt you if you mistakenly think bb equals the product of the constants in the factors, but that's actually what gives us the last term (20). Choice (D) claims b=20b = 20, which is the constant term, not the coefficient of xx. Remember this pattern: when you have (x+m)(x+n)(x + m)(x + n), the coefficient of xx in the expanded form is always m+nm + n. Here, 4+5=94 + 5 = 9, so the answer is (B). This shortcut can save you time—you don't always need to fully expand when you're just looking for the middle coefficient.

Question 15

What is the complete factorization of x4−81x^4 - 81?

  1. (x2+9)(x+3)(x−3)(x^2 + 9)(x + 3)(x - 3) (correct answer)
  2. (x+3)2(x−3)2(x + 3)^2(x - 3)^2
  3. (x2+9)(x2−9)(x^2 + 9)(x^2 - 9)
  4. (x+9)(x−9)(x + 9)(x - 9)
Explanation: When you encounter an expression like x4−81x^4 - 81, recognize this as a difference of squares pattern. The key insight is that both x4x^4 and 8181 are perfect squares: x4=(x2)2x^4 = (x^2)^2 and 81=9281 = 9^2. Start by applying the difference of squares formula a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b): x4−81=(x2)2−92=(x2+9)(x2−9)x^4 - 81 = (x^2)^2 - 9^2 = (x^2 + 9)(x^2 - 9) However, you're not done yet! Look at the second factor: x2−9x^2 - 9 is also a difference of squares since x2=(x)2x^2 = (x)^2 and 9=329 = 3^2. Factor this further: x2−9=(x+3)(x−3)x^2 - 9 = (x + 3)(x - 3) The first factor x2+9x^2 + 9 cannot be factored further using real numbers (it's a sum of squares). Therefore, the complete factorization is (x2+9)(x+3)(x−3)(x^2 + 9)(x + 3)(x - 3), which is choice A. Looking at the wrong answers: Choice B gives (x+3)2(x−3)2(x + 3)^2(x - 3)^2, which would expand to (x2−9)2=x4−18x2+81(x^2 - 9)^2 = x^4 - 18x^2 + 81, not our original expression. Choice C stops at the first factorization step and isn't complete. Choice D treats the expression as if it were x2−81x^2 - 81 instead of x4−81x^4 - 81. Remember: Always check if your factored terms can be factored further. Difference of squares problems often require multiple steps, so don't stop after the first factorization!

Question 16

A garden's area is A(x)=x2+7x+12A(x)=x^2+7x+12. Which represents the factored form of A(x)A(x)?

  1. (x+3)(x+4)(x+3)(x+4) (correct answer)
  2. (x−3)(x−4)(x-3)(x-4)
  3. x(x+7)+12x(x+7)+12
  4. (x+2)(x+6)(x+2)(x+6)
Explanation: This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring quadratic expressions. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions, solving equations, and understanding polynomial functions. For the given expression x² + 7x + 12, the correct factors are derived by identifying pairs of numbers that multiply to give the constant term (12) and add to give the linear coefficient (7). Choice A, (x+3)(x+4), is correct because 3 × 4 = 12 and 3 + 4 = 7, and when expanded gives x² + 4x + 3x + 12 = x² + 7x + 12. Choice B is incorrect because it would expand to x² - 7x + 12, showing a sign error that students make when they don't carefully check whether factors should be positive or negative. Teaching strategies include using the FOIL method to verify answers, creating factor trees for the constant term, and emphasizing the relationship between the signs in the original expression and the factored form.

Question 17

Which of the following expressions is completely factored?

  1. x(x2−4)x(x^2 - 4)
  2. 3(x2+2x−8)3(x^2 + 2x - 8)
  3. 2x(x+3)(x−3)2x(x + 3)(x - 3) (correct answer)
  4. (x2+1)(x2−9)(x^2 + 1)(x^2 - 9)
Explanation: When you see "completely factored," you need to check whether each expression has been broken down into its simplest possible factors—meaning no factor can be factored further. Let's examine each option systematically. In choice C, 2x(x+3)(x−3)2x(x + 3)(x - 3), we have three factors: 2x2x, (x+3)(x + 3), and (x−3)(x - 3). The factor 2x2x contains the prime number 2 and the variable xx, neither of which can be factored further. The linear factors (x+3)(x + 3) and (x−3)(x - 3) are also in their simplest form. This expression is completely factored. Choice A, x(x2−4)x(x^2 - 4), looks factored but isn't complete. The factor (x2−4)(x^2 - 4) is a difference of squares that can be factored further as (x+2)(x−2)(x + 2)(x - 2). So this should be x(x+2)(x−2)x(x + 2)(x - 2). Choice B, 3(x2+2x−8)3(x^2 + 2x - 8), has the constant 3 factored out, but the quadratic x2+2x−8x^2 + 2x - 8 can be factored as (x+4)(x−2)(x + 4)(x - 2). The complete factorization would be 3(x+4)(x−2)3(x + 4)(x - 2). Choice D, (x2+1)(x2−9)(x^2 + 1)(x^2 - 9), contains (x2−9)(x^2 - 9), which is another difference of squares that factors as (x+3)(x−3)(x + 3)(x - 3). The factor (x2+1)(x^2 + 1) cannot be factored further over the real numbers. Study tip: Always check for difference of squares patterns (a2−b2a^2 - b^2) and factorable quadratics when verifying complete factorization. If you can factor any piece further, it's not completely factored.

Question 18

If 2x2−8x+k2x^2 - 8x + k can be factored as 2(x−a)22(x - a)^2 for some constants aa and kk, what is the value of kk?

  1. k=4k = 4
  2. k=8k = 8 (correct answer)
  3. k=12k = 12
  4. k=16k = 16
Explanation: When you see a quadratic expression that can be factored in a specific form, you're working with completing the square or perfect square trinomials. The key insight is that both expressions must be equivalent, so their expanded forms must match exactly. Let's expand the given factored form 2(x−a)22(x - a)^2 and compare it to 2x2−8x+k2x^2 - 8x + k. First, expand (x−a)2=x2−2ax+a2(x - a)^2 = x^2 - 2ax + a^2, so 2(x−a)2=2x2−4ax+2a22(x - a)^2 = 2x^2 - 4ax + 2a^2. Now we can match coefficients between 2x2−8x+k2x^2 - 8x + k and 2x2−4ax+2a22x^2 - 4ax + 2a^2:
  • The x2x^2 coefficients already match: 2=22 = 2 ✓
  • The xx coefficients: −8=−4a-8 = -4a, so a=2a = 2
  • The constant terms: k=2a2=2(2)2=2(4)=8k = 2a^2 = 2(2)^2 = 2(4) = 8
Therefore, k=8k = 8, which is choice B. Let's check why the other answers are wrong. Choice A (k=4k = 4) likely comes from incorrectly calculating a2a^2 instead of 2a22a^2. Choice C (k=12k = 12) might result from adding a2+2a=4+8a^2 + 2a = 4 + 8. Choice D (k=16k = 16) could come from using 4a24a^2 instead of 2a22a^2. Strategy tip: When matching factored forms to expanded forms, always expand the factored version completely and match coefficients term by term. This systematic approach prevents calculation errors and ensures you don't miss any factors.

Question 19

Which of the following expressions cannot be factored using integer coefficients?

  1. x2+5x+6x^2 + 5x + 6
  2. x2+x−6x^2 + x - 6
  3. x2+5x+7x^2 + 5x + 7 (correct answer)
  4. x2−x−6x^2 - x - 6
Explanation: When you encounter quadratic expressions like these, you're being tested on factoring—specifically whether a quadratic can be written as a product of two binomials with integer coefficients. For an expression x2+bx+cx^2 + bx + c, you need two integers that multiply to give cc and add to give bb. Let's check each expression systematically. For choice A, x2+5x+6x^2 + 5x + 6, you need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so this factors as (x+2)(x+3)(x + 2)(x + 3). For choice B, x2+x−6x^2 + x - 6, you need two numbers that multiply to -6 and add to 1. Those are 3 and -2, giving (x+3)(x−2)(x + 3)(x - 2). Choice D, x2−x−6x^2 - x - 6, requires two numbers that multiply to -6 and add to -1. Those are -3 and 2, so it factors as (x−3)(x+2)(x - 3)(x + 2). However, choice C, x2+5x+7x^2 + 5x + 7, requires two integers that multiply to 7 and add to 5. Since 7 is prime, its only integer factor pairs are 1×7 and (-1)×(-7). The pair 1 and 7 adds to 8, while -1 and -7 adds to -8. Neither gives us the required sum of 5, making this expression impossible to factor using integer coefficients. The key strategy: when checking if a quadratic factors with integer coefficients, list all factor pairs of the constant term and see if any pair sums to the middle coefficient. If none work, it can't be factored over the integers.

Question 20

What is the complete factorization of 5x2−455x^2 - 45?

  1. 5(x2−9)5(x^2 - 9)
  2. 5(x−3)25(x - 3)^2
  3. 5(x+3)(x−3)5(x + 3)(x - 3) (correct answer)
  4. (5x+15)(x−3)(5x + 15)(x - 3)
Explanation: When you see an expression like 5x2−455x^2 - 45, you're looking at a polynomial that can be factored by first identifying the greatest common factor (GCF), then checking if what remains follows a special factoring pattern. Start by factoring out the GCF. Both terms share a factor of 5: 5x2−45=5(x2−9)5x^2 - 45 = 5(x^2 - 9). Now examine what's inside the parentheses. The expression x2−9x^2 - 9 is a difference of squares since x2−9=x2−32x^2 - 9 = x^2 - 3^2. The difference of squares pattern a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b) gives us x2−9=(x+3)(x−3)x^2 - 9 = (x + 3)(x - 3). Therefore, the complete factorization is 5(x+3)(x−3)5(x + 3)(x - 3), which is answer C. Let's examine why the other choices fall short. Choice A, 5(x2−9)5(x^2 - 9), correctly factors out the GCF but stops there without recognizing the difference of squares pattern—this is incomplete factoring. Choice B, 5(x−3)25(x - 3)^2, represents a perfect square trinomial, but if you expand it you get 5(x2−6x+9)=5x2−30x+455(x^2 - 6x + 9) = 5x^2 - 30x + 45, which doesn't match our original expression. Choice D, (5x+15)(x−3)(5x + 15)(x - 3), expands to 5x2−15x+15x−45=5x2−455x^2 - 15x + 15x - 45 = 5x^2 - 45, which seems correct until you notice that 5x+155x + 15 can be factored further as 5(x+3)5(x + 3), making this factorization incomplete. Remember: complete factorization means factoring until no further factoring is possible. Always check for GCF first, then look for special patterns like difference of squares.