Math 3 Quiz: Remainder Theorem

11 questions · exam conditions

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Remainder TheoremQuestion 1 of 11

When polynomial $P(x) = x^4 - 3x^3 + ax^2 + bx - 8$ is divided by $(x^2 - 4)$ , the remainder is $2x + 6$ . What is the value of $P(2) + P(-2)$ ?

P(2) + P(-2) = 8

P(2) + P(-2) = 12

P(2) + P(-2) = 16

P(2) + P(-2) = 20

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Math 3 Quiz

Math 3 Quiz: Remainder Theorem

Practice Remainder Theorem in Math 3 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 Remainder Theorem, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 3.

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

When polynomial $P(x) = x^4 - 3x^3 + ax^2 + bx - 8$ is divided by $(x^2 - 4)$ , the remainder is $2x + 6$ . What is the value of $P(2) + P(-2)$ ?

$P(2) + P(-2) = 8$
$P(2) + P(-2) = 12$
$P(2) + P(-2) = 16$ (correct answer)
$P(2) + P(-2) = 20$

Explanation: When P(x) is divided by (x² - 4) = (x-2)(x+2), the remainder is 2x + 6. By the division algorithm, P(x) = (x² - 4)Q(x) + 2x + 6 for some polynomial Q(x). Therefore, P(2) = (4-4)Q(2) + 2(2) + 6 = 0 + 4 + 6 = 10, and P(-2) = (4-4)Q(-2) + 2(-2) + 6 = 0 - 4 + 6 = 2. Thus P(2) + P(-2) = 10 + 2 = 16. Choice A results from calculation errors. Choice B incorrectly assumes both values are 6. Choice D incorrectly adds the constant term of the remainder twice.

Question 2

If $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ where $n \geq 2$ , and $P(2) = 15$ , $P(3) = 15$ , what can be concluded about the remainder when $P(x) - 15$ is divided by $(x-2)(x-3)$ ?

The remainder is $15$
The remainder is $0$ (correct answer)
The remainder is $x - 2$
The remainder cannot be determined without knowing the degree $n$

Explanation: Let Q(x) = P(x) - 15. Then Q(2) = P(2) - 15 = 15 - 15 = 0 and Q(3) = P(3) - 15 = 15 - 15 = 0. Since Q(x) has zeros at x = 2 and x = 3, both (x-2) and (x-3) are factors of Q(x). Therefore, (x-2)(x-3) is a factor of Q(x), which means the remainder when Q(x) is divided by (x-2)(x-3) is 0. Choice A confuses the function value with the remainder. Choice C gives an arbitrary linear expression. Choice D incorrectly suggests that the degree affects this conclusion when we have sufficient information from the Remainder Theorem.

Question 3

If $P(x) = 2x^4 - 3x^3 + 5x^2 - 7x + 1$ and $P(2) = -5$ , what is the remainder when $P(x)$ is divided by $(x - 2)$ ?

The remainder is $-5$ (correct answer)
The remainder is $5$
The remainder cannot be determined without performing polynomial long division
The remainder is $0$ since $P(2) eq 0$

Explanation: By the Remainder Theorem, when a polynomial P(x) is divided by (x - a), the remainder equals P(a). Since P(2) = -5, the remainder when P(x) is divided by (x - 2) is -5. Choice B has the wrong sign. Choice C reflects a misunderstanding that the Remainder Theorem provides a shortcut to finding remainders. Choice D incorrectly suggests that a non-zero function value means zero remainder.

Question 4

If $g(x) = x^3 - 4x^2 + mx + n$ and $g(x)$ is divisible by $(x - 2)$ , which statement about $g(2)$ is correct?

$g(2) = 0$ regardless of the values of $m$ and $n$
$g(2) = 2m + n - 8$ for any values of $m$ and $n$
$g(2)$ cannot equal zero unless $m = 0$ and $n = 0$
$g(2) = 0$ only if $2m + n = 8$ (correct answer)

Explanation: If g(x) is divisible by (x - 2), then by the Remainder Theorem, the remainder when dividing by (x - 2) is g(2) = 0. Computing: g(2) = 8 - 16 + 2m + n = 2m + n - 8. For g(2) = 0, we need 2m + n - 8 = 0, so 2m + n = 8. Choice A ignores that m and n affect g(2). Choice B gives the general form but doesn't use the divisibility condition. Choice C is incorrect about when g(2) can be zero.

Question 5

A polynomial $f(x)$ satisfies $f(x+2) = x^3 + 6x^2 + 11x + 6$ . If $g(x) = f(x) - 3$ , what is the remainder when $g(x)$ is divided by $(x+1)$ ?

The remainder is $-3$ (correct answer)
The remainder is $-1$
The remainder is $0$
The remainder is $3$

Explanation: First, find f(x). Since f(x+2) = x³ + 6x² + 11x + 6, let u = x + 2, so x = u - 2. Then f(u) = (u-2)³ + 6(u-2)² + 11(u-2) + 6. Expanding: (u-2)³ = u³ - 6u² + 12u - 8, 6(u-2)² = 6(u² - 4u + 4) = 6u² - 24u + 24, 11(u-2) = 11u - 22. So f(u) = u³ - 6u² + 12u - 8 + 6u² - 24u + 24 + 11u - 22 + 6 = u³ + (12-24+11)u + (-8+24-22+6) = u³ - u + 0 = u³ - u. Therefore, f(x) = x³ - x. Now g(x) = f(x) - 3 = x³ - x - 3. The remainder when g(x) is divided by (x+1) is g(-1) = (-1)³ - (-1) - 3 = -1 + 1 - 3 = -3. Choice B, C, and D represent incorrect calculations of g(-1).

Question 6

A polynomial $P(x)$ of degree 4 satisfies $P(0) = 5$ , $P(1) = 3$ , $P(-1) = 7$ , $P(2) = 1$ , and $P(-2) = 9$ . What is the remainder when $P(x)$ is divided by $(x^2-1)$ ?

$2x + 5$
$-2x + 5$ (correct answer)
$x + 4$
$-x + 4$

Explanation: When P(x) is divided by x² - 1 = (x-1)(x+1), the remainder must be a linear polynomial of the form ax + b. By the division algorithm, P(x) = (x² - 1)Q(x) + ax + b. At the roots of x² - 1, we have P(1) = a(1) + b = a + b and P(-1) = a(-1) + b = -a + b. From the given conditions: P(1) = 3, so a + b = 3. P(-1) = 7, so -a + b = 7. Adding these equations: 2b = 10, so b = 5. Subtracting the second from the first: 2a = -4, so a = -2. Therefore, the remainder is -2x + 5. Choice A has the wrong sign for the coefficient of x. Choice C and D have incorrect constant terms.

Question 7

A polynomial $f(x)$ satisfies $f(3) = 7$ and $f(-2) = 12$ . If $g(x) = f(x+1) - 5$ , what is the remainder when $g(x)$ is divided by $(x-2)$ ?

The remainder is $2$ (correct answer)
The remainder is $7$
The remainder is $12$
The remainder is $17$

Explanation: To find the remainder when g(x) is divided by (x-2), we need to find g(2). Since g(x) = f(x+1) - 5, we have g(2) = f(2+1) - 5 = f(3) - 5 = 7 - 5 = 2. Therefore, the remainder is 2. Choice B incorrectly uses f(3) directly. Choice C incorrectly uses f(-2). Choice D incorrectly adds f(3) + f(-2) - 5.

Question 8

When the polynomial $h(x) = x^4 + ax^3 + bx^2 + cx + d$ is divided by $(x^2 - 1)$ , the remainder is $3x + 5$ . What is $h(1) + h(-1)$ ?

$8$
$10$ (correct answer)
$6$
$16$

Explanation: When h(x) is divided by (x² - 1) = (x - 1)(x + 1), the remainder is 3x + 5. By the division algorithm, h(x) = (x² - 1)q(x) + 3x + 5 for some quotient q(x). At x = 1: h(1) = (1 - 1)q(1) + 3(1) + 5 = 0 + 8 = 8. At x = -1: h(-1) = (1 - 1)q(-1) + 3(-1) + 5 = 0 - 3 + 5 = 2. Therefore, h(1) + h(-1) = 8 + 2 = 10. Choice A gives h(1) only. Choice C gives h(-1) + h(1) with a computational error. Choice D doubles the sum incorrectly.

Question 9

If $f(x) = x^5 - 2x^3 + x^2 - 4$ and $f(a) = 0$ for some value $a$ , which statement must be true?

The remainder when $f(x)$ is divided by $(x - a)$ equals $-4$
The remainder when $f(x)$ is divided by $(x - a)$ equals $0$ (correct answer)
$(x - a)$ is a factor of $f(x) + 4$
$f(x)$ has no other zeros besides $a$

Explanation: By the Remainder Theorem, when f(x) is divided by (x - a), the remainder is f(a). Since f(a) = 0, the remainder is 0. This also means (x - a) is a factor of f(x). Choice A incorrectly uses the constant term. Choice C is incorrect because if f(a) = 0, then f(a) + 4 = 4 ≠ 0, so (x - a) is not a factor of f(x) + 4. Choice D makes an unjustified claim about other zeros.

Question 10

A polynomial $f(x)$ leaves a remainder of $8$ when divided by $(x - 3)$ and a remainder of $-2$ when divided by $(x + 1)$ . What is the value of $f(3) - f(-1)$ ?

$6$
$10$ (correct answer)
$-10$
Cannot be determined from the given information

Explanation: By the Remainder Theorem, when f(x) is divided by (x - 3), the remainder is f(3) = 8. When f(x) is divided by (x + 1) = (x - (-1)), the remainder is f(-1) = -2. Therefore, f(3) - f(-1) = 8 - (-2) = 10. Choice A results from computing 8 - 2 = 6 (sign error). Choice C gives -(f(3) - f(-1)). Choice D reflects uncertainty about whether the Remainder Theorem provides sufficient information.

Question 11

Consider the polynomial $s(x) = x^4 - 5x^3 + 7x^2 - 3x + 2$ . If $s(x)$ is divided by $(x - c)$ where $c > 3$ , and the remainder is negative, which inequality must $c$ satisfy?

$3 < c < 4$
$c > 4$
$s(c) < 0$ for the specific value of $c$ (correct answer)
No such $c > 3$ exists since $s(x) > 0$ for all $x > 3$

Explanation: By the Remainder Theorem, when s(x) is divided by (x - c), the remainder equals s(c). For the remainder to be negative, we need s(c) < 0. This is the direct application of the Remainder Theorem. Choice A assumes a specific interval without justification. Choice B makes an unfounded claim about c > 4. Choice D incorrectly claims s(x) is always positive for x > 3, which would require analyzing the behavior of this quartic polynomial.