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Remainder Theorem - Math 3

  1. If P(x)=2x4−3x3+5x2−7x+1P(x) = 2x^4 - 3x^3 + 5x^2 - 7x + 1 and P(2)=−5P(2) = -5, what is the remainder when P(x)P(x) is divided by (x−2)(x - 2)?
    Answer: The remainder is −5-5 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.
  2. If f(x)=x5−2x3+x2−4f(x) = x^5 - 2x^3 + x^2 - 4 and f(a)=0f(a) = 0 for some value aa, which statement must be true?
    Answer: The remainder when f(x)f(x) is divided by (x−a)(x - a) equals 00 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.
  3. A polynomial f(x)f(x) satisfies f(x+2)=x3+6x2+11x+6f(x+2) = x^3 + 6x^2 + 11x + 6. If g(x)=f(x)−3g(x) = f(x) - 3, what is the remainder when g(x)g(x) is divided by (x+1)(x+1)?
    Answer: The remainder is −3-3 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).
  4. If g(x)=x3−4x2+mx+ng(x) = x^3 - 4x^2 + mx + n and g(x)g(x) is divisible by (x−2)(x - 2), which statement about g(2)g(2) is correct?
    Answer: g(2)=0g(2) = 0 only if 2m+n=82m + n = 8 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.
  5. When polynomial P(x)=x4−3x3+ax2+bx−8P(x) = x^4 - 3x^3 + ax^2 + bx - 8 is divided by (x2−4)(x^2 - 4), the remainder is 2x+62x + 6. What is the value of P(2)+P(−2)P(2) + P(-2)?
    Answer: P(2)+P(−2)=16P(2) + P(-2) = 16 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.
  6. A polynomial f(x)f(x) leaves a remainder of 88 when divided by (x−3)(x - 3) and a remainder of −2-2 when divided by (x+1)(x + 1). What is the value of f(3)−f(−1)f(3) - f(-1)?
    Answer: 1010 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.
  7. When the polynomial h(x)=x4+ax3+bx2+cx+dh(x) = x^4 + ax^3 + bx^2 + cx + d is divided by (x2−1)(x^2 - 1), the remainder is 3x+53x + 5. What is h(1)+h(−1)h(1) + h(-1)?
    Answer: 1010 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.
  8. A polynomial f(x)f(x) satisfies f(3)=7f(3) = 7 and f(−2)=12f(-2) = 12. If g(x)=f(x+1)−5g(x) = f(x+1) - 5, what is the remainder when g(x)g(x) is divided by (x−2)(x-2)?
    Answer: The remainder is 22 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.
  9. Consider the polynomial s(x)=x4−5x3+7x2−3x+2s(x) = x^4 - 5x^3 + 7x^2 - 3x + 2. If s(x)s(x) is divided by (x−c)(x - c) where c>3c > 3, and the remainder is negative, which inequality must cc satisfy?
    Answer: s(c)<0s(c) < 0 for the specific value of cc 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.
  10. A polynomial P(x)P(x) of degree 4 satisfies P(0)=5P(0) = 5, P(1)=3P(1) = 3, P(−1)=7P(-1) = 7, P(2)=1P(2) = 1, and P(−2)=9P(-2) = 9. What is the remainder when P(x)P(x) is divided by (x2−1)(x^2-1)?
    Answer: −2x+5-2x + 5 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.
  11. If P(x)=xn+an−1xn−1+⋯+a1x+a0P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 where n≥2n \geq 2, and P(2)=15P(2) = 15, P(3)=15P(3) = 15, what can be concluded about the remainder when P(x)−15P(x) - 15 is divided by (x−2)(x−3)(x-2)(x-3)?
    Answer: The remainder is 00 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.

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