ACT Math Quiz: Polynomial Functions

What this quiz covers

This quiz focuses on Polynomial Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

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.

Question 1

1. $3x^2 + 6x + 3$
2. $3x^2 - 2x + 1$
3. $3x^2 - 6x + 3$ (correct answer)
4. $3x^2 - 2x + 3$

Explanation: This tests the distributive property: a factor outside parentheses multiplies every term inside, and it keeps each term's sign. Multiplying term by term gives $3 \cdot x^2 = 3x^2$, $3 \cdot (-2x) = -6x$, and $3 \cdot 1 = 3$, so the equivalent polynomial is $3x^2 - 6x + 3$. The version $3x^2 + 6x + 3$ comes from dropping the minus sign on the middle term, $3x^2 - 2x + 1$ comes from multiplying only the leading term and leaving the rest untouched, and $3x^2 - 2x + 3$ comes from distributing to the first and last terms but forgetting the middle one. When a single factor sits outside parentheses, count the terms inside first and make sure your answer shows that same number of products, each with its original sign.

Question 2

1. $6$
2. $1$
3. $2$ (correct answer)
4. $4$

Explanation: The degree of a polynomial is the largest exponent on the variable, and coefficients play no role in it. In $g(x) = -4x^2 + 6x - 1$ the exponents are $2$, $1$, and $0$, so the largest is $2$ and the degree is $2$. The answer $4$ mistakes the coefficient $-4$ for an exponent and $6$ mistakes the coefficient $6$ the same way, while $1$ picks the exponent of the middle term instead of the highest one. Circle only the exponents before comparing them, and remember that a constant term carries an invisible exponent of $0$.

Question 3

1. 7 (correct answer)
2. -5
3. 2
4. -1

Explanation: We need to identify the leading coefficient of 7x³ - 5x² + 2x - 1. The leading coefficient is the coefficient of the term with the highest degree. The highest degree term is 7x³, which has degree 3. The coefficient of this term is 7, so the leading coefficient is 7.

Question 4

1. $y = (x + 3)^2 + 2$
2. $y = (x - 3)^2 + 2$ (correct answer)
3. $y = (x - 3)^2 - 2$
4. $y = (x - 2)^2 + 3$

Explanation: This is a graph transformations question testing vertex form. Choice B (y = (x − 3)² + 2) is correct — shifting a parabola 3 units to the right replaces x with (x − 3): counterintuitively, a rightward shift subtracts from x. Shifting 2 units up adds 2 outside the squared term. Result: y = (x − 3)² + 2. Choice A (y = (x + 3)² + 2) shifts the graph LEFT 3 units, not right — adding inside the parentheses moves the vertex to x = −3, which is a leftward shift. Choice C (y = (x − 3)² − 2) correctly shifts right 3 but shifts DOWN 2 units instead of up. Choice D (y = (x − 2)² + 3) swaps the shift values — moving right 2 and up 3 instead of right 3 and up 2. Pro tip: Horizontal shifts in vertex form are counterintuitive: (x − h) shifts the graph h units to the RIGHT, and (x + h) shifts it LEFT. Vertical shifts are straightforward: adding outside moves up, subtracting moves down.

Question 5

1. $x = -3$
2. $x = 3$
3. $x = 5$ (correct answer)
4. $y = 1$

Explanation: This is a rational functions question testing the difference between holes and vertical asymptotes. Choice C (x = 5) is correct — factor both expressions: numerator = (x − 3)(x + 3); denominator = (x − 5)(x + 3). The (x + 3) factor cancels, creating a removable discontinuity (hole) at x = −3. The remaining denominator factor (x − 5) sets to zero at x = 5, creating the vertical asymptote. Choice A (x = −3) identifies a zero of the denominator, but since (x + 3) cancels from both numerator and denominator, it produces a hole — not an asymptote. Choice B (x = 3) identifies a zero of the numerator — but zeros of the numerator create x-intercepts, not asymptotes. Choice D (y = 1) correctly identifies the horizontal asymptote (leading coefficients both equal 1), but the question asks for the vertical asymptote. Pro tip: Vertical asymptotes occur where denominator factors equal zero AFTER canceling any shared factors with the numerator. Always factor and cancel first — shared factors produce holes, not asymptotes.

Question 6

1. 42 (correct answer)
2. 37
3. 28
4. 30

Explanation: We need to evaluate f(-5) for f(x) = x² - 3x + 2. Substitute x = -5: f(-5) = (-5)² - 3(-5) + 2. Calculate each term carefully: 25 + 15 + 2 = 42. The middle term becomes positive because we have -3 times a negative number. Choice B might result from sign errors with the linear term.

Question 7

1. $x^2 + x + 2$ (correct answer)
2. $x^2 + x - 2$
3. $x^2 + x + 3$
4. $x^2 - x + 2$

Explanation: We need to simplify x(x - 1) + 2(x + 1) by distributing and combining like terms. First distribute: x(x - 1) = x² - x and 2(x + 1) = 2x + 2. Combine: x² - x + 2x + 2 = x² + (-x + 2x) + 2 = x² + x + 2. Choice B would result from sign errors when combining the x terms.

Question 8

1. 1
2. 2
3. 3 (correct answer)
4. 4

Explanation: The degree of a polynomial is the highest power of the variable. In the polynomial 4x³ - x + 2, we identify the highest exponent among all terms. The terms have powers 3, 1, and 0 respectively, so the highest power is 3. Choice D might confuse the leading coefficient (4) with the degree.

Question 9

1. 1
2. 2 (correct answer)
3. 3
4. 0

Explanation: The degree of a polynomial is the highest power of the variable. In the polynomial x² + 2x + 1, we identify the highest exponent among all terms. The terms have powers 2, 1, and 0 respectively, so the highest power is 2. Choice C might confuse the number of terms with the degree.

Question 10

1. -6
2. 5
3. -3
4. 2 (correct answer)

Explanation: The leading coefficient is the coefficient of the term with the highest degree. In the polynomial 2x³ - 3x² + 5x - 6, the highest degree term is 2x³ with degree 3. The coefficient of this term is 2, which is the leading coefficient. Choice C might incorrectly identify the coefficient of the x² term instead.

Question 11

1. $\{-1,5\}$ (correct answer)
2. $\{1,-5\}$
3. $\{-5,-1\}$
4. $\{0,5\}$

Explanation: To find zeros of f(x) = x² - 4x - 5, we set f(x) = 0 and solve x² - 4x - 5 = 0. Factoring: we need two numbers that multiply to -5 and add to -4, which are -5 and 1. So (x - 5)(x + 1) = 0, giving x = 5 or x = -1. The zeros are {-1, 5}. Choice B incorrectly has {1, -5}, reversing the signs.

Question 12

1. 1
2. 2 (correct answer)
3. 3
4. 0

Explanation: We need to find the degree of the polynomial 5x² - 2x. The degree is the highest power of x in the expression. Looking at each term: 5x² has degree 2, and -2x has degree 1. Therefore, the degree of the polynomial is 2.

Question 13

1. $x^2 + 2x$
2. $x^2 - 2x + 1$
3. $x^2 + 1$
4. $x^2 + 2x + 1$ (correct answer)

Explanation: We need to expand (x + 1)² using the perfect square formula. Using (a + b)² = a² + 2ab + b², where a = x and b = 1: (x + 1)² = x² + 2(x)(1) + 1² = x² + 2x + 1.

Question 14

1. $9$
2. $-1$
3. $4$ (correct answer)
4. $2$

Explanation: The leading coefficient is the coefficient of the term with the highest degree. In r(x) = 4x² - x + 9, identify the highest degree term: 4x² has degree 2, which is the highest among all terms. The coefficient of this term is 4, making it the leading coefficient. Choice A might confuse the constant term with the leading coefficient.

Question 15

1. $12$
2. $-5$ (correct answer)
3. $5$
4. $-12$

Explanation: The leading coefficient is the coefficient of the term with the highest degree. In b(x) = -5x + 12, this is a linear polynomial where -5x has degree 1 (the highest degree) and 12 has degree 0. The coefficient of the highest degree term -5x is -5, making it the leading coefficient. Choice C might incorrectly ignore the negative sign.

Question 16

1. 1
2. 2
3. 3 (correct answer)
4. 0

Explanation: The degree of a polynomial is the highest power of the variable. In the polynomial x³ - 4x + 2, we identify the highest exponent among all terms. The terms have powers 3, 1, and 0 respectively, so the highest power is 3. Choice B might confuse the coefficient of the linear term with the degree.

Question 17

1. $4x^2-1$
2. $4x^2-4x+1$ (correct answer)
3. $2x^2-4x+1$
4. $4x^2+4x+1$

Explanation: To expand (2x - 1)², we use the formula (a - b)² = a² - 2ab + b². Here a = 2x and b = 1, so: (2x)² - 2(2x)(1) + 1² = 4x² - 4x + 1. We can verify by FOIL: (2x - 1)(2x - 1) = 4x² - 2x - 2x + 1 = 4x² - 4x + 1. Choice D shows 4x² + 4x + 1, which has the wrong sign on the middle term.

Question 18

1. 0 (correct answer)
2. 3
3. -3
4. 6

Explanation: We need to evaluate f(3) for f(x) = x² - 4x + 3. Substitute x = 3: f(3) = (3)² - 4(3) + 3. Calculate each term: 9 - 12 + 3 = 0. This shows that x = 3 is a root of this polynomial since f(3) = 0. Choice B might result from forgetting to include the constant term.

Question 19

1. 3
2. 0
3. 7 (correct answer)
4. 10

Explanation: We need to evaluate f(0) by substituting x = 0 into f(x) = 3x³ + 2x² - x + 7. When x = 0, all terms with x become zero: f(0) = 3(0)³ + 2(0)² - (0) + 7 = 0 + 0 - 0 + 7 = 7. The constant term gives us the value directly.

Question 20

1. 1
2. 2
3. 3 (correct answer)
4. 4

Explanation: We need to find the degree of the polynomial x³ + x² + x + 1. The degree is the highest power of x that appears in the expression. Looking at each term: x³ has degree 3, x² has degree 2, x has degree 1, and 1 has degree 0. Therefore, the degree of the polynomial is 3.