ACT Math · Learn by Concept

ACT Math Help: Quadratics And Polynomials

Review real example questions for Quadratics And Polynomials in ACT Math.

Question 1 / 10

0 of 10 answered

To expand an expression used in a perimeter calculation, compute: What is $(x+5)(x-2)$ ?

All questions

Question 1

To expand an expression used in a perimeter calculation, compute: What is $(x+5)(x-2)$ ?

$x^2+3x+10$
$x^2-3x-10$
$x^2+3x-10$ (correct answer)
$x^2+7x-10$

Explanation: Expanding a product of two binomials means multiplying every term of the first by every term of the second, which FOIL organizes as First, Outer, Inner, Last. For $(x+5)(x-2)$ the First product is $x^2$ , the Outer is $-2x$ , the Inner is $5x$ , and the Last is $(5)(-2)=-10$ , so combining the two middle terms gives $x^2+3x-10$ . The result $x^2+3x+10$ keeps the correct middle term but multiplies $5$ and $-2$ as if both were positive, and $x^2-3x-10$ reverses which middle term is negative, giving $-5x+2x$ instead of $5x-2x$ . The result $x^2+7x-10$ comes from adding $5$ and $2$ for the middle coefficient instead of combining $+5x$ with $-2x$ . When you expand binomials, compute the constant product with its signs first, then add the two cross terms rather than adding the raw numbers, since the sign on each cross term decides the middle coefficient.

Question 2

The polynomial $p(x) = x^3 - 4x^2 - 7x + 10$ has $(x - 1)$ as a factor. What are all the other zeros of $p(x)$ ?

$-5$ and $2$
$5$ and $-2$ (correct answer)
$5$ and $2$
$-5$ and $-2$

Explanation: This is a polynomial factors and zeros question testing synthetic division and factoring. Choice B (5 and −2) is correct — since (x − 1) is a factor, divide p(x) by (x − 1) using synthetic division: coefficients 1, −4, −7, 10 divided by root 1: bring down 1; 1×1 − 4 = −3; −3×1 − 7 = −10; −10×1 + 10 = 0 ✓. Quotient: x² − 3x − 10 = (x − 5)(x + 2). Zeros: x = 5 and x = −2. Choice A (−5 and 2) flips both signs of the correct zeros. Choice C (5 and 2) gets x = 5 correct but uses x = 2 instead of x = −2, likely from factoring x² − 3x − 10 as (x − 5)(x − 2) (wrong sign on the second factor). Choice D (−5 and −2) negates both correct zeros. Pro tip: After dividing out the known factor, you're left with a simpler polynomial to factor. Here: (x − 1)(x² − 3x − 10) = 0. Factor x² − 3x − 10: find two numbers that multiply to −10 and sum to −3: −5 and +2. So (x − 5)(x + 2) = 0 → x = 5 or x = −2.

Question 3

Which of the following is equivalent to $(5x^2 - 3x + 7) - (2x^2 + 4x - 1)$ ?

$3x^2 - 7x + 8$ (correct answer)
$3x^2 + x + 6$
$7x^2 + x + 6$
$7x^2 - 7x + 8$

Explanation: This is a polynomial operations question testing distribution of a negative sign. Choice A (3x² − 7x + 8) is correct — distribute the negative sign: (5x² − 3x + 7) − (2x² + 4x − 1) = 5x² − 3x + 7 − 2x² − 4x + 1. Key step: −(−1) = +1. Combine: (5−2)x² + (−3−4)x + (7+1) = 3x² − 7x + 8. Choice B (3x² + x + 6) correctly subtracts x² terms but fails to distribute the negative on the x term: −3x − (4x) treated as −3x + 4x = +x, and −(−1) = −1 instead of +1. Choice C (7x² + x + 6) adds x² terms instead of subtracting (5 + 2 = 7) and makes both sign errors. Choice D (7x² − 7x + 8) adds x² terms (5 + 2 = 7) but correctly handles x and constant terms. Pro tip: When subtracting a polynomial, distribute the negative sign to EVERY term inside the parentheses. Rewrite −(2x² + 4x − 1) as (−2x² − 4x + 1) before combining. The term that trips most students is −(−1) = +1.

Question 4

A ball's height is modeled by $h(t)=t^2-5t+6$ . To find when it hits the ground, solve $t^2-5t+6=0$ .

$t=\dfrac{5\pm\sqrt{1}}{2}$
$t=2$ and $t=3$ (correct answer)
$t=-2$ and $t=-3$
$t=1$ and $t=6$

Explanation: To solve $t^2 - 5t + 6 = 0$ , factor by finding two numbers that multiply to 6 and add to -5. Since $(-2)(-3) = 6$ and $(-2) + (-3) = -5$ , we get $(t - 2)(t - 3) = 0$ . Setting each factor equal to zero gives $t = 2$ and $t = 3$ . Choice A uses the quadratic formula unnecessarily, and choices C and D give incorrect values.

Question 5

What is the value of $f(3)$ for $f(x) = x^2 - 4x + 4$ ?

1 (correct answer)
0
4
3

Explanation: To find $f(3)$ for $f(x) = x^2 - 4x + 4$ , we substitute $x = 3$ into the function. We get $f(3) = (3)^2 - 4(3) + 4 = 9 - 12 + 4 = 1$ . Notice that this function can also be written as $(x - 2)^2$ , so $f(3) = (3 - 2)^2 = 1^2 = 1$ .

Question 6

What is the value of $f(0)$ for $f(x) = 3x^2 - 4x + 7$ ?

7 (correct answer)
3
0
4

Explanation: To find $f(0)$ for $f(x) = 3x^2 - 4x + 7$ , we substitute $x = 0$ into the function. We get $f(0) = 3(0)^2 - 4(0) + 7 = 0 - 0 + 7 = 7$ . When evaluating polynomial functions at zero, all terms with $x$ become zero, leaving only the constant term.

Question 7

Factor: $x^2 - 10x + 21$

$(x + 4)(x + 6)$
$(x + 3)(x + 7)$
$(x - 4)(x - 6)$
$(x - 3)(x - 7)$ (correct answer)

Explanation: To factor x² - 10x + 21, we need two numbers that multiply to 21 and add to -10. The numbers are -3 and -7 since (-3)(-7) = 21 and -3 + (-7) = -10. Therefore, x² - 10x + 21 = (x - 3)(x - 7). Choice C has factors that multiply to 24, not 21.

Question 8

Which of the following lists all the solutions to the quadratic equation $2x^2 - 5x - 12 = 0$ ?

$-4$ and $\frac{3}{2}$
$-\frac{3}{2}$ and $4$ (correct answer)
$-3$ and $2$
$3$ and $-2$

Explanation: The correct answer is B (−3/2 and 4). Factor 2x² − 5x − 12 = 0. Look for factors of (2)(−12) = −24 that add to −5: those are −8 and +3. Rewrite: 2x² − 8x + 3x − 12 = 0 → 2x(x − 4) + 3(x − 4) = 0 → (2x + 3)(x − 4) = 0. Solutions: x = −3/2 and x = 4. A (−4 and 3/2) flips the signs of both solutions. C (−3 and 2) ignores the leading coefficient of 2, incorrectly factoring as (x + 3)(x − 2) = 0. D (3 and −2) also ignores the leading 2, factoring as (x − 3)(x + 2) = 0. When the leading coefficient isn't 1, always account for it in the factoring process.

Question 9

The polynomial $P(x) = x^3 - 2x^2 - 5x + 6$ has $(x - 1)$ as a factor. What are the other two roots of $P(x) = 0$ ?

$x = 2, x = 3$
$x = -2, x = 3$ (correct answer)
$x = 2, x = -3$
$x = -2, x = -3$

Explanation: When a polynomial has a given factor, you can find the other factors by polynomial division or by testing the roots. Since (x - 1) is a factor, x = 1 is a root. To find the other roots, divide P(x) by (x - 1) using synthetic division with 1: you get x² - x - 6, which factors as (x - 3)(x + 2). Therefore, the complete factorization is P(x) = (x - 1)(x - 3)(x + 2), giving roots x = 1, 3, -2. The question asks for the "other two roots" (besides x = 1), which are -2 and 3.

Question 10

A rectangular garden has a length that is 3 feet longer than its width. If the perimeter of the garden is 54 feet, what is the area of the garden, in square feet?

180 (correct answer)
182
200
729

Explanation: This is a perimeter and area question requiring setting up and solving an equation. Choice A (180) is correct — let w = width, then length = w + 3. Perimeter: 2(w) + 2(w + 3) = 54 → 4w + 6 = 54 → 4w = 48 → w = 12. Length = 15. Area = 12 × 15 = 180 sq ft. Choice B (182) results from an off-by-one error in the dimensions, using 13 × 14 = 182 — likely from a perimeter setup error that produces w = 13. Choice C (200) may come from a perimeter equation error that yields dimensions like 10 × 20. Choice D (729) results from treating the half-perimeter (27) as both dimensions of a square and squaring it: 27² = 729. Pro tip: Set up the perimeter equation explicitly — don't try to guess dimensions. Write 2w + 2(w + 3) = 54, solve for w, then compute the area from the actual dimensions.