mathematics beginner 10 sample questions

Polynomials MCQ Practice Test

Multi-term expressions and operations

Q1. Given (x^3 + 2x^2 - 4x - 8) / (x + 2) = 0, find the value of x.

A. x = -2
B. x = 4
C. x = -1
D. x = 2 ✓

Explanation: The polynomial (x^3 + 2x^2 - 4x - 8) / (x + 2) = 0 can be simplified. Factoring: x^3 + 2x^2 - 4x - 8 = (x + 2)(x^2 - 4) = (x + 2)^2(x - 2). Dividing by (x + 2) gives (x + 2)(x - 2) = 0, so x = 2 or x = -2. However, x = -2 is excluded from the domain because it makes the denominator zero. Therefore, the only valid solution is x = 2.

Q2. If the polynomial p(x) = 2x^3 + 3x^2 - 4x + 1 has roots r1, r2 and r3, what is the value of r1 · r2 · r3?

A. -1/2 ✓
B. 1/2
C. -3/2
D. 3/2

Explanation: By Vieta's formulas, for ax^3 + bx^2 + cx + d the product of the roots is -d/a. Here r1·r2·r3 = -1/2.

Q3. If f(x) = 2x^3 + 5x^2 - 3x + 1 and g(x) = x^2 + 2x + 1, what is the value of f(g(2))?

A. 1837 ✓
B. 1683
C. 2011
D. 2118

Explanation: g(2) = 2^2 + 2·2 + 1 = 9. Then f(9) = 2(729) + 5(81) - 3(9) + 1 = 1458 + 405 - 27 + 1 = 1837.

Q4. Given the polynomial f(x) = x^3 - 2x^2 - 5x + 6, what is f(-2)?

A. 0 ✓
B. -10
C. 10
D. -14

Explanation: Substituting x = -2: f(-2) = (-2)^3 - 2(-2)^2 - 5(-2) + 6 = -8 - 8 + 10 + 6 = 0. The result confirms that (x + 2) is a factor — indeed x^3 - 2x^2 - 5x + 6 = (x - 1)(x + 2)(x - 3).

Q5. Given the polynomial f(x) = x^3 + 4x^2 - 3x - 9, find the value of f(-2) using the Remainder Theorem.

A. The remainder is 17
B. The remainder is -1
C. The remainder is 1
D. The remainder is 5 ✓

Explanation: The Remainder Theorem: the remainder of dividing f(x) by (x + 2) equals f(-2). f(-2) = (-2)^3 + 4(-2)^2 - 3(-2) - 9 = -8 + 16 + 6 - 9 = 5.

Q6. Given the polynomial f(x) = x^3 + 2x^2 - 7x - 12, what is the value of f(-3)?

A. f(-3) = 0 ✓
B. f(-3) = -14
C. f(-3) = 14
D. f(-3) = 2

Explanation: Substituting: f(-3) = (-27) + 18 + 21 - 12 = 0. A zero value means (x + 3) is a factor, and indeed f(x) = (x + 3)(x^2 - x - 4). Note the old premise 'f(-2) = 0' was wrong - f(-2) = 2.

Q7. Find the value of k such that the polynomial f(x) = x^3 - x^2 - 9x + k has a root that is also a root of the polynomial g(x) = x^2 - 9.

A. k = 9 ✓
B. k = 7
C. k = 11
D. k = 13

Explanation: g(x) = x^2 - 9 has roots x = 3 and x = -3. For a shared root, f(3) = 27 - 9 - 27 + k = 0 gives k = 9; checking, f(-3) = -27 - 9 + 27 + 9 = 0, so with k = 9 both roots of g are roots of f. The original pairing with g(x) = x^2 + 3x - 6 produced irrational roots and k of roughly 0.51, matching no option.

Q8. If \( p(x) = x^2 - 6x - 16 \), what is the value of \( p(2x) \) when \( x = 3 \)?

A. 6
B. 12
C. 24
D. 48

Explanation: To find p(2x) when x = 3: first compute 2x = 2(3) = 6. Then p(6) = 6^2 - 6(6) - 16 = 36 - 36 - 16 = -16. Therefore p(2x) when x = 3 equals -16.

Q9. Given the polynomial f(x) = x^3 - 6x^2 + 11x - 6, what is the value of k if f(x) = (x - 1)(x^2 + kx + 6)?

A. k = -5 ✓
B. k = 5
C. k = -1
D. k = 1

Explanation: Expanding: (x-1)(x^2+kx+6) = x^3 + (k-1)x^2 + (6-k)x - 6. Matching the x^2 coefficient: k-1 = -6, so k = -5; check with the x coefficient: 6-(-5) = 11 ✓.

Q10. Given the polynomial f(x) = 2x^3 + 5x^2 - 7x - 3, what is the remainder when f(x) is divided by (x + 2)?

A. -3
B. 5
C. 15 ✓
D. 23

Explanation: By the Remainder Theorem, when f(x) = 2x^3 + 5x^2 - 7x - 3 is divided by (x + 2), the remainder is f(-2) = 2(-8) + 5(4) - 7(-2) - 3 = -16 + 20 + 14 - 3 = 15. The remainder must be a constant (degree less than the divisor).

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