ACCUPLACER Advanced Algebra & Functions Quiz: Polynomial Arithmetic
20 questions · exam conditions
0:00
Polynomial ArithmeticQuestion 1 of 20

If R(x)=4x3−2x2+7x−1R(x) = 4x^3 - 2x^2 + 7x - 1 and S(x)=x3+3x2−5x+6S(x) = x^3 + 3x^2 - 5x + 6, what is the constant term in R(x)+S(x)R(x) + S(x)?

-7
-5
7
5
← Back to quizzes

ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Polynomial Arithmetic

Practice Polynomial Arithmetic in ACCUPLACER Advanced Algebra & Functions 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 Polynomial Arithmetic, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Advanced Algebra & Functions.

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

If R(x)=4x3−2x2+7x−1R(x) = 4x^3 - 2x^2 + 7x - 1 and S(x)=x3+3x2−5x+6S(x) = x^3 + 3x^2 - 5x + 6, what is the constant term in R(x)+S(x)R(x) + S(x)?

  1. -7
  2. -5
  3. 7
  4. 5 (correct answer)
Explanation: When you see polynomial addition problems, focus on combining like terms—terms with the same variable and exponent. The constant term is the term without any variables, which will be the easiest to identify. To find R(x)+S(x)R(x) + S(x), you add corresponding terms from each polynomial: R(x)+S(x)=(4x3−2x2+7x−1)+(x3+3x2−5x+6)R(x) + S(x) = (4x^3 - 2x^2 + 7x - 1) + (x^3 + 3x^2 - 5x + 6) Group like terms together:
  • x3x^3 terms: 4x3+x3=5x34x^3 + x^3 = 5x^3
  • x2x^2 terms: −2x2+3x2=x2-2x^2 + 3x^2 = x^2
  • xx terms: 7x+(−5x)=2x7x + (-5x) = 2x
  • Constant terms: −1+6=5-1 + 6 = 5
So R(x)+S(x)=5x3+x2+2x+5R(x) + S(x) = 5x^3 + x^2 + 2x + 5, making the constant term 5, which is choice D. Let's examine the wrong answers: Choice A (-7) likely comes from incorrectly subtracting the constant terms instead of adding: −1−6=−7-1 - 6 = -7. Choice B (-5) might result from adding the xx coefficients incorrectly: 7+(−5)=27 + (-5) = 2, but then confusing this with the constant calculation. Choice C (7) could come from forgetting the negative sign on the first constant: 1+6=71 + 6 = 7. Strategy tip: When adding polynomials, organize your work by writing like terms in columns or explicitly grouping them as shown above. This prevents sign errors and ensures you don't accidentally combine unlike terms. Always double-check your arithmetic with constant terms since they're often where careless mistakes happen.

Question 2

Which of the following is the result of subtracting (2x2−7x+3)(2x^2 - 7x + 3) from (−5x2+4x−8)(-5x^2 + 4x - 8)?

  1. −7x2+11x−11-7x^2 + 11x - 11 (correct answer)
  2. −7x2−3x−5-7x^2 - 3x - 5
  3. −3x2−3x−5-3x^2 - 3x - 5
  4. 7x2−11x+117x^2 - 11x + 11
Explanation: To subtract (2x2−7x+3)(2x^2 - 7x + 3) from (−5x2+4x−8)(-5x^2 + 4x - 8), set up the expression: (−5x2+4x−8)−(2x2−7x+3)(-5x^2 + 4x - 8) - (2x^2 - 7x + 3). Distribute the negative sign to each term in the second polynomial: −5x2+4x−8−2x2+7x−3-5x^2 + 4x - 8 - 2x^2 + 7x - 3. Combine like terms: (−5x2−2x2)+(4x+7x)+(−8−3)=−7x2+11x−11(-5x^2 - 2x^2) + (4x + 7x) + (-8 - 3) = -7x^2 + 11x - 11.

Question 3

Which expression is the result of the multiplication (7a3−4b2)(7a3+4b2)(7a^3 - 4b^2)(7a^3 + 4b^2)?

  1. 49a6+16b449a^6 + 16b^4
  2. 49a9−16b449a^9 - 16b^4
  3. 49a6−16b449a^6 - 16b^4 (correct answer)
  4. 49a6−56a3b2−16b449a^6 - 56a^3b^2 - 16b^4
Explanation: This is a product of the sum and difference of two terms, which follows the pattern (x−y)(x+y)=x2−y2(x - y)(x + y) = x^2 - y^2. Here, x=7a3x = 7a^3 and y=4b2y = 4b^2. So, the result is (7a3)2−(4b2)2(7a^3)^2 - (4b^2)^2. Squaring each part gives 49a(3∗2)−16b(2∗2)=49a6−16b449a^{(3*2)} - 16b^{(2*2)} = 49a^6 - 16b^4.

Question 4

If P(x)=2x2−x+5P(x) = 2x^2 - x + 5 and Q(x)=3x2+2x−1Q(x) = 3x^2 + 2x - 1, what is 2P(x)−Q(x)2P(x) - Q(x)?

  1. x2−4x+9x^2 - 4x + 9
  2. x2−4x+11x^2 - 4x + 11 (correct answer)
  3. x2+9x^2 + 9
  4. 7x2+97x^2 + 9
Explanation: First, find 2P(x)2P(x) by multiplying each term of P(x)P(x) by 2: 2(2x2−x+5)=4x2−2x+102(2x^2 - x + 5) = 4x^2 - 2x + 10. Then, subtract Q(x)Q(x) from this result: (4x2−2x+10)−(3x2+2x−1)(4x^2 - 2x + 10) - (3x^2 + 2x - 1). Distribute the negative: 4x2−2x+10−3x2−2x+14x^2 - 2x + 10 - 3x^2 - 2x + 1. Combine like terms: (4x2−3x2)+(−2x−2x)+(10+1)=x2−4x+11(4x^2 - 3x^2) + (-2x - 2x) + (10 + 1) = x^2 - 4x + 11.

Question 5

Which of the following is equivalent to (x−3)(x+3)(2x−1)(x - 3)(x + 3)(2x - 1)?

  1. 2x3−x2−18x+92x^3 - x^2 - 18x + 9 (correct answer)
  2. 2x3−18x−92x^3 - 18x - 9
  3. 2x3−x2−6x+92x^3 - x^2 - 6x + 9
  4. 2x3−13x2+24x−92x^3 - 13x^2 + 24x - 9
Explanation: First, multiply the first two binomials, which are a difference of squares: (x−3)(x+3)=x2−9(x - 3)(x + 3) = x^2 - 9. Then, multiply this result by the third binomial: (x2−9)(2x−1)(x^2 - 9)(2x - 1). Using distribution (FOIL): (x2)(2x)+(x2)(−1)+(−9)(2x)+(−9)(−1)=2x3−x2−18x+9(x^2)(2x) + (x^2)(-1) + (-9)(2x) + (-9)(-1) = 2x^3 - x^2 - 18x + 9.

Question 6

A box has a length of (x+4)(x+4), a width of (x−1)(x-1), and a height of (x)(x). Which polynomial represents the volume of the box?

  1. x3−4xx^3 - 4x
  2. x3+3x2−4xx^3 + 3x^2 - 4x (correct answer)
  3. x3+3x−4x^3 + 3x - 4
  4. 3x+33x + 3
Explanation: The volume of a box is length × width × height. So, the volume is (x+4)(x−1)(x)(x+4)(x-1)(x). First, multiply the two binomials: (x+4)(x−1)=x2−x+4x−4=x2+3x−4(x+4)(x-1) = x^2 - x + 4x - 4 = x^2 + 3x - 4. Then, multiply this result by the height, xx: x(x2+3x−4)=x3+3x2−4xx(x^2 + 3x - 4) = x^3 + 3x^2 - 4x.

Question 7

Which expression results from simplifying (3a2−2ab+5b2)−(a2−6ab−3b2)(3a^2 - 2ab + 5b^2) - (a^2 - 6ab - 3b^2)?

  1. 2a2−8ab+2b22a^2 - 8ab + 2b^2
  2. 2a2+4ab+2b22a^2 + 4ab + 2b^2
  3. 4a2−8ab+2b24a^2 - 8ab + 2b^2
  4. 2a2+4ab+8b22a^2 + 4ab + 8b^2 (correct answer)
Explanation: To subtract the polynomials, distribute the negative sign to each term in the second polynomial: 3a2−2ab+5b2−a2+6ab+3b23a^2 - 2ab + 5b^2 - a^2 + 6ab + 3b^2. Then, group and combine like terms: (3a2−a2)+(−2ab+6ab)+(5b2+3b2)(3a^2 - a^2) + (-2ab + 6ab) + (5b^2 + 3b^2). This simplifies to 2a2+4ab+8b22a^2 + 4ab + 8b^2.

Question 8

The expression (ax2−5x+2)−(3x2+bx−4)(ax^2 - 5x + 2) - (3x^2 + bx - 4) is simplified to −x2+8x+c-x^2 + 8x + c. What is the value of a−ba - b?

  1. −11-11
  2. 55
  3. 1515 (correct answer)
  4. 1717
Explanation: First, simplify the given expression: ax2−5x+2−3x2−bx+4=(a−3)x2+(−5−b)x+6ax^2 - 5x + 2 - 3x^2 - bx + 4 = (a-3)x^2 + (-5-b)x + 6. Now, equate the coefficients of this simplified form with −x2+8x+c-x^2 + 8x + c. For the x2x^2 term: a−3=−1a - 3 = -1, which means a=2a = 2. For the xx term: −5−b=8-5 - b = 8, which means −b=13-b = 13 or b=−13b = -13. The question asks for a−ba - b. Substitute the values: 2−(−13)=2+13=152 - (-13) = 2 + 13 = 15.

Question 9

Let P(x)=4x2−3x+1P(x) = 4x^2 - 3x + 1, Q(x)=x2+5x−6Q(x) = x^2 + 5x - 6, and R(x)=−2x2+xR(x) = -2x^2 + x. Which of the following is equivalent to P(x)−Q(x)+R(x)P(x) - Q(x) + R(x)?

  1. x2+3x−5x^2 + 3x - 5
  2. x2−7x+7x^2 - 7x + 7 (correct answer)
  3. 3x2+3x−53x^2 + 3x - 5
  4. 5x2−9x+75x^2 - 9x + 7
Explanation: Substitute the given polynomials into the expression: (4x2−3x+1)−(x2+5x−6)+(−2x2+x)(4x^2 - 3x + 1) - (x^2 + 5x - 6) + (-2x^2 + x). Distribute the negative sign to Q(x)Q(x): 4x2−3x+1−x2−5x+6−2x2+x4x^2 - 3x + 1 - x^2 - 5x + 6 - 2x^2 + x. Group and combine like terms: (4x2−x2−2x2)+(−3x−5x+x)+(1+6)=x2−7x+7(4x^2 - x^2 - 2x^2) + (-3x - 5x + x) + (1 + 6) = x^2 - 7x + 7.

Question 10

A company's weekly revenue, RR, from selling xx units of a product is given by the polynomial R(x)=50x−0.1x2R(x) = 50x - 0.1x^2. The weekly cost, CC, to produce xx units is given by C(x)=500+10xC(x) = 500 + 10x. Which polynomial represents the weekly profit, P(x)=R(x)−C(x)P(x) = R(x) - C(x)?

  1. −0.1x2+40x−500-0.1x^2 + 40x - 500 (correct answer)
  2. −0.1x2+40x+500-0.1x^2 + 40x + 500
  3. −0.1x2+60x+500-0.1x^2 + 60x + 500
  4. 0.1x2−40x+5000.1x^2 - 40x + 500
Explanation: Profit is calculated as Revenue minus Cost, so P(x)=R(x)−C(x)P(x) = R(x) - C(x). Substitute the given polynomials: P(x)=(50x−0.1x2)−(500+10x)P(x) = (50x - 0.1x^2) - (500 + 10x). Distribute the negative sign to the cost polynomial: 50x−0.1x2−500−10x50x - 0.1x^2 - 500 - 10x. Combine like terms and write in standard form (by decreasing power of x): −0.1x2+(50x−10x)−500=−0.1x2+40x−500-0.1x^2 + (50x - 10x) - 500 = -0.1x^2 + 40x - 500.

Question 11

A rectangular garden has a length of (3x+5)(3x + 5) meters and a width of (2x−1)(2x - 1) meters. Inside the garden, there is a square fountain with a side length of (x−2)(x - 2) meters. Which polynomial represents the area of the garden not covered by the fountain?

  1. 5x2+3x−15x^2 + 3x - 1
  2. 5x2+7x−15x^2 + 7x - 1
  3. 5x2+11x−95x^2 + 11x - 9 (correct answer)
  4. 7x2+3x−17x^2 + 3x - 1
Explanation: The area of the garden is (3x+5)(2x−1)=6x2−3x+10x−5=6x2+7x−5(3x + 5)(2x - 1) = 6x^2 - 3x + 10x - 5 = 6x^2 + 7x - 5. The area of the fountain is (x−2)2=x2−4x+4(x - 2)^2 = x^2 - 4x + 4. To find the remaining area, subtract the fountain's area from the garden's area: (6x2+7x−5)−(x2−4x+4)=6x2+7x−5−x2+4x−4=5x2+11x−9(6x^2 + 7x - 5) - (x^2 - 4x + 4) = 6x^2 + 7x - 5 - x^2 + 4x - 4 = 5x^2 + 11x - 9.

Question 12

When (2x−3)(x2+5x−4)(2x - 3)(x^2 + 5x - 4) is expanded and simplified, what is the coefficient of x2x^2?

  1. -3
  2. 7 (correct answer)
  3. 10
  4. -23
Explanation: Using the distributive property: (2x−3)(x2+5x−4)=2x(x2+5x−4)−3(x2+5x−4)=2x3+10x2−8x−3x2−15x+12=2x3+7x2−23x+12(2x - 3)(x^2 + 5x - 4) = 2x(x^2 + 5x - 4) - 3(x^2 + 5x - 4) = 2x^3 + 10x^2 - 8x - 3x^2 - 15x + 12 = 2x^3 + 7x^2 - 23x + 12. The coefficient of x2x^2 is 77. Choice A is the coefficient of −3x2-3x^2 alone. Choice C is the coefficient of 10x210x^2 alone. Choice D is the coefficient of xx.

Question 13

What is the degree of the polynomial that results from (x3−2x+1)(x2+3x−4)+(x4−2x5)(x^3 - 2x + 1)(x^2 + 3x - 4) + (x^4 - 2x^5)?

  1. 4
  2. 5 (correct answer)
  3. 6
  4. 8
Explanation: The product (x3−2x+1)(x2+3x−4)(x^3 - 2x + 1)(x^2 + 3x - 4) has highest degree term x3⋅x2=x5x^3 \cdot x^2 = x^5 with coefficient 1. When we add (x4−2x5)(x^4 - 2x^5), the x5x^5 terms combine as 1x5+(−2x5)=−x51x^5 + (-2x^5) = -x^5. Since this doesn't equal zero, the degree remains 5. Choice A incorrectly assumes the x5x^5 terms cancel completely. Choice C incorrectly adds the degrees 3+2+1=63 + 2 + 1 = 6. Choice D incorrectly multiplies the degrees.

Question 14

Which of the following is equivalent to (3x4+2)(3x4−2)(3x^4 + 2)(3x^4 - 2)?

  1. 9x8−49x^8 - 4 (correct answer)
  2. 9x8+49x^8 + 4
  3. 9x6−49x^6 - 4
  4. 3x8−43x^8 - 4
Explanation: This multiplication is in the form of a difference of squares, (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2, where a=3x4a = 3x^4 and b=2b = 2. We calculate a2=(3x4)2=32(x4)2=9x8a^2 = (3x^4)^2 = 3^2(x^4)^2 = 9x^8 and b2=22=4b^2 = 2^2 = 4. The result is 9x8−49x^8 - 4.

Question 15

What is the simplified form of the expression 4x2−[x(3x−1)−(2x2−5x)]4x^2 - [x(3x - 1) - (2x^2 - 5x)]?

  1. −x2+6x-x^2 + 6x
  2. 3x2+4x3x^2 + 4x
  3. 3x2+6x3x^2 + 6x
  4. 3x2−4x3x^2 - 4x (correct answer)
Explanation: First, simplify the expression inside the brackets. Distribute xx: 3x2−x3x^2 - x. Distribute the negative: −(2x2−5x)=−2x2+5x-(2x^2 - 5x) = -2x^2 + 5x. Combine these results inside the bracket: [3x2−x−2x2+5x]=[x2+4x][3x^2 - x - 2x^2 + 5x] = [x^2 + 4x]. Now, substitute this back into the original expression: 4x2−[x2+4x]4x^2 - [x^2 + 4x]. Distribute the outer negative sign: 4x2−x2−4x4x^2 - x^2 - 4x. Combine like terms to get 3x2−4x3x^2 - 4x.

Question 16

What is the coefficient of the x2x^2 term in the simplified expansion of (2x2−5x+3)(x−4)(2x^2 - 5x + 3)(x - 4)?

  1. −13-13 (correct answer)
  2. −8-8
  3. −5-5
  4. −3-3
Explanation: To find the x2x^2 term, we only need to find the products of terms that result in x2x^2. These are: (the x2x^2 term from the first polynomial) × (the constant term from the second) AND (the xx term from the first) × (the xx term from the second). First product: (2x2)(−4)=−8x2(2x^2)(-4) = -8x^2. Second product: (−5x)(x)=−5x2(-5x)(x) = -5x^2. Add these results: −8x2+(−5x2)=−13x2-8x^2 + (-5x^2) = -13x^2. The coefficient is -13.

Question 17

Which of the following is equivalent to the expression (2x+1)2−(2x−1)2(2x + 1)^2 - (2x - 1)^2?

  1. 00
  2. 22
  3. 8x2+28x^2 + 2
  4. 8x8x (correct answer)
Explanation: First, expand each squared binomial. (2x+1)2=(2x)2+2(2x)(1)+12=4x2+4x+1(2x + 1)^2 = (2x)^2 + 2(2x)(1) + 1^2 = 4x^2 + 4x + 1. (2x−1)2=(2x)2−2(2x)(1)+12=4x2−4x+1(2x - 1)^2 = (2x)^2 - 2(2x)(1) + 1^2 = 4x^2 - 4x + 1. Now, subtract the second result from the first: (4x2+4x+1)−(4x2−4x+1)(4x^2 + 4x + 1) - (4x^2 - 4x + 1). Distribute the negative: 4x2+4x+1−4x2+4x−14x^2 + 4x + 1 - 4x^2 + 4x - 1. The 4x24x^2 terms cancel and the 11 terms cancel, leaving 4x+4x=8x4x + 4x = 8x.

Question 18

The sum of two polynomials is 6x2−10x+56x^2 - 10x + 5. If one of the polynomials is −2x2+3x−7-2x^2 + 3x - 7, what is the other polynomial?

  1. 4x2−7x−24x^2 - 7x - 2
  2. 8x2−7x−28x^2 - 7x - 2
  3. 8x2−13x+128x^2 - 13x + 12 (correct answer)
  4. −8x2+13x−12-8x^2 + 13x - 12
Explanation: To find the other polynomial, subtract the known polynomial from the sum: (6x2−10x+5)−(−2x2+3x−7)(6x^2 - 10x + 5) - (-2x^2 + 3x - 7). Distribute the negative sign to the second polynomial: 6x2−10x+5+2x2−3x+76x^2 - 10x + 5 + 2x^2 - 3x + 7. Combine like terms: (6x2+2x2)+(−10x−3x)+(5+7)=8x2−13x+12(6x^2 + 2x^2) + (-10x - 3x) + (5 + 7) = 8x^2 - 13x + 12.

Question 19

Which of the following is equivalent to (4x3−3y)2(4x^3 - 3y)^2?

  1. 16x6−9y216x^6 - 9y^2
  2. 16x5−24x3y+9y216x^5 - 24x^3y + 9y^2
  3. 16x6−12x3y+9y216x^6 - 12x^3y + 9y^2
  4. 16x6−24x3y+9y216x^6 - 24x^3y + 9y^2 (correct answer)
Explanation: Use the formula for the square of a binomial, (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, with a=4x3a = 4x^3 and b=3yb = 3y. a2=(4x3)2=16x6a^2 = (4x^3)^2 = 16x^6. 2ab=2(4x3)(3y)=24x3y2ab = 2(4x^3)(3y) = 24x^3y. b2=(3y)2=9y2b^2 = (3y)^2 = 9y^2. Combining these gives 16x6−24x3y+9y216x^6 - 24x^3y + 9y^2.

Question 20

Simplify the expression: 5x2−(2x+3)(x−4)5x^2 - (2x + 3)(x - 4).

  1. 3x2−5x−123x^2 - 5x - 12
  2. 3x2+5x−123x^2 + 5x - 12
  3. 3x2+5x+123x^2 + 5x + 12 (correct answer)
  4. 3x2+11x+123x^2 + 11x + 12
Explanation: First, multiply the binomials: (2x+3)(x−4)=2x2−8x+3x−12=2x2−5x−12(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12. Now substitute this back into the original expression: 5x2−(2x2−5x−12)5x^2 - (2x^2 - 5x - 12). Distribute the negative sign: 5x2−2x2+5x+125x^2 - 2x^2 + 5x + 12. Combine like terms: 3x2+5x+123x^2 + 5x + 12.