Question 1

Simplify: (x + 3)² − (x − 3)². What is the result?

Accepted Answer

Step 1: Expand (x + 3)² = x² + 6x + 9. Step 2: Expand (x − 3)² = x² − 6x + 9. Step 3: Subtract: (x² + 6x + 9) − (x² − 6x + 9) = x² + 6x + 9 − x² + 6x − 9 = 12x. Therefore, the result is 12x.

Question 2

If the polynomial x² − 5x + m has roots α and β such that α + β = 5 and αβ = 6, find the value of m.

Accepted Answer

Step 1: Use Vieta's formulas for a quadratic x² + bx + c with roots α and β: α + β = −b and αβ = c. Step 2: For x² − 5x + m, we have −b = −(−5) = 5 and c = m. Step 3: Given αβ = 6, we have m = 6. Therefore, m = 6.

Question 3

What is the degree of the polynomial q(x) = 7x⁴ − 3x² + 2x + 9?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial q(x) = 7x⁴ − 3x² + 2x + 9. Step 2: The highest power is x⁴, so the degree is 4. Therefore, the answer is 4.

Question 4

If r(x) = x² − 6x + 8, what are the roots of the polynomial?

Accepted Answer

Step 1: Factor r(x) = x² − 6x + 8. Step 2: Find two numbers that multiply to 8 and add to −6: these are −2 and −4. Step 3: r(x) = (x − 2)(x − 4). Step 4: The roots are x = 2 and x = 4. Therefore, the answer is 2 and 4.

Question 5

What is the degree of the polynomial 4x⁵ − 3x³ + 2x − 7?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial. Step 2: The terms are 4x⁵, −3x³, 2x, and −7. Step 3: The highest power is 5. Therefore, the degree is 5.

Question 6

Simplify: (x + 3)(x − 2) − (x + 1)²

Accepted Answer

Step 1: Expand (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6. Step 2: Expand (x + 1)² = x² + 2x + 1. Step 3: Subtract: (x² + x − 6) − (x² + 2x + 1) = x² + x − 6 − x² − 2x − 1 = −x − 7. Therefore, the answer is −x − 7.

Question 7

Divide the polynomial 2x³ + 7x² + 2x − 3 by (x + 1). What is the remainder?

Accepted Answer

Step 1: Use the Remainder Theorem: when p(x) is divided by (x − a), the remainder is p(a). Step 2: Here, we divide by (x + 1), so a = −1. Step 3: p(−1) = 2(−1)³ + 7(−1)² + 2(−1) − 3 = −2 + 7 − 2 − 3 = 0. Therefore, the remainder is 0.

Question 8

If (x + 2) is a factor of the polynomial x² + kx + 6, find the value of k.

Accepted Answer

Step 1: If (x + 2) is a factor, then x = −2 is a root of the polynomial. Step 2: Substitute x = −2 into x² + kx + 6 = 0. Step 3: (−2)² + k(−2) + 6 = 0 → 4 − 2k + 6 = 0 → 10 − 2k = 0 → 2k = 10 → k = 5. Therefore, k = 5.

Question 9

If p(x) = 2x³ + 5x² - 3x + 4, then find the value of p(2).

Accepted Answer

Step 1: We need to substitute x = 2 into the polynomial p(x) = 2x³ + 5x² - 3x + 4.
Step 2: p(2) = 2(2)³ + 5(2)² - 3(2) + 4
Step 3: p(2) = 2(8) + 5(4) - 6 + 4
Step 4: p(2) = 16 + 20 - 6 + 4
Step 5: p(2) = 34
Therefore, p(2) = 34.

Question 10

If p(x) = x³ - 6x² + 11x - 6, find the sum of all roots of the polynomial.

Accepted Answer

Step 1: For a cubic polynomial ax³ + bx² + cx + d, the sum of roots = -b/a. Step 2: Here, a = 1, b = -6. Step 3: Sum of roots = -(-6)/1 = 6. Therefore, the answer is 6.

Question 11

If the polynomial r(x) = x³ - 4x² - 7x + 10 has roots α, β, and γ, find α + β + γ using Vieta's formulas.

Accepted Answer

Step 1: For a cubic polynomial x³ + ax² + bx + c, Vieta's formula states that the sum of roots = -a. Step 2: In r(x) = x³ - 4x² - 7x + 10, the coefficient of x² is -4. Step 3: Therefore, α + β + γ = -(-4) = 4.

Question 12

If p(x) = x⁴ - 5x² + 4, find the number of real roots of p(x) = 0.

Accepted Answer

Step 1: Let y = x². Then p(x) = y² - 5y + 4. Step 2: Factor: y² - 5y + 4 = (y - 1)(y - 4) = 0. Step 3: So y = 1 or y = 4. Step 4: If x² = 1, then x = ±1 (two real roots). Step 5: If x² = 4, then x = ±2 (two real roots). Step 6: Total real roots = 4.

Question 13

If (x + 3) is a factor of the polynomial q(x) = x³ + 6x² + 9x + k, find the value of k.

Accepted Answer

Step 1: If (x + 3) is a factor, then q(−3) = 0. Step 2: Substitute x = −3: q(−3) = (−3)³ + 6(−3)² + 9(−3) + k = −27 + 54 − 27 + k = 0. Step 3: Simplify: 0 + k = 0, so k = 0.

Question 14

The polynomial p(x) = x² - 7x + 12 can be factored as (x - a)(x - b). What is the value of a + b?

Accepted Answer

Step 1: Factor x² - 7x + 12 by finding two numbers that multiply to 12 and add to -7. Step 2: Those numbers are -3 and -4 (since -3 × -4 = 12 and -3 + -4 = -7). Step 3: So p(x) = (x - 3)(x - 4), meaning a = 3 and b = 4. Step 4: Therefore, a + b = 3 + 4 = 7.

Question 15

If p(x) = x³ – 6x² + 11x – 6, then which of the following is a factor of p(x)?

Accepted Answer

Step 1: Test x = 1: p(1) = 1 – 6 + 11 – 6 = 0. So (x – 1) is a factor. Step 2: Test x = 2: p(2) = 8 – 24 + 22 – 6 = 0. So (x – 2) is a factor. Step 3: Test x = 3: p(3) = 27 – 54 + 33 – 6 = 0. So (x – 3) is a factor. Step 4: The polynomial factors as (x–1)(x–2)(x–3). Among the options, (x – 3) is a valid factor. Therefore, answer is (x – 3).

Question 16

If p(x) = x³ − 6x² + 11x − 6 is a polynomial, and (x − 1) is one of its factors, find the sum of the remaining two roots.

Accepted Answer

Step 1: Since (x − 1) is a factor, x = 1 is a root. Verify: p(1) = 1 − 6 + 11 − 6 = 0 ✓. Step 2: By Vieta's formulas, for p(x) = x³ − 6x² + 11x − 6, the sum of all three roots = −(−6)/1 = 6. Step 3: If one root is 1, then the sum of the remaining two roots = 6 − 1 = 5.

Question 17

The polynomial s(x) = 2x³ + 3x² - 8x - 12 can be factored by grouping. What is one of its linear factors?

Accepted Answer

Step 1: Group terms: s(x) = (2x³ + 3x²) + (-8x - 12). Step 2: Factor each group: = x²(2x + 3) - 4(2x + 3). Step 3: Factor out (2x + 3): = (2x + 3)(x² - 4). Step 4: Factor x² - 4 as a difference of squares: = (2x + 3)(x - 2)(x + 2). Step 5: One linear factor is (x - 2), (x + 2), or (2x + 3).

Question 18

If (x - 2) is a factor of the polynomial q(x) = x³ + ax² - 5x + 6, find the value of a.

Accepted Answer

Step 1: If (x - 2) is a factor, then q(2) = 0. Step 2: Substitute x = 2: (2)³ + a(2)² - 5(2) + 6 = 0. Step 3: 8 + 4a - 10 + 6 = 0. Step 4: 4a + 4 = 0. Step 5: a = -1. Therefore, the answer is -1.

Question 19

If the polynomial r(x) = 2x³ - 3x² + bx - 4 leaves a remainder of 5 when divided by (x - 1), find b.

Accepted Answer

Step 1: By the Remainder Theorem, when r(x) is divided by (x - 1), the remainder equals r(1). Step 2: r(1) = 2(1)³ - 3(1)² + b(1) - 4 = 5. Step 3: 2 - 3 + b - 4 = 5. Step 4: b - 5 = 5. Step 5: b = 10. Therefore, the answer is 10.

Question 20

If p(x) = x³ - 6x² + 11x - 6 and q(x) = x² - 3x + 2, then the remainder when p(x) is divided by q(x) is:

Accepted Answer

Step 1: Use polynomial long division or remainder theorem. Since q(x) = x² - 3x + 2 is quadratic, the remainder must be linear: R(x) = ax + b. Step 2: Write p(x) = q(x)·d(x) + R(x), where d(x) is the quotient. Step 3: Perform division: x³ - 6x² + 11x - 6 = (x² - 3x + 2)(x - 3) + 0. Step 4: The remainder is 0.

Question 21

If the polynomial x³ + px² + qx + r is divisible by (x-1)(x+2), then which of the following is true?

Accepted Answer

Step 1: If p(x) is divisible by (x-1)(x+2), then p(1) = 0 and p(-2) = 0. Step 2: p(1) = 1 + p + q + r = 0 → p + q + r = -1. Step 3: p(-2) = -8 + 4p - 2q + r = 0 → 4p - 2q + r = 8. Step 4: From equation 1: r = -1 - p - q. Step 5: Substitute into equation 2: 4p - 2q + (-1 - p - q) = 8 → 3p - 3q - 1 = 8 → 3p - 3q = 9 → p - q = 3. Step 6: Therefore p = q + 3.

Question 22

The polynomial p(x) = x⁴ + ax³ + bx² + cx + d has roots 1, 2, 3, and 4. If p(0) = 24, then the value of a + b + c is:

Accepted Answer

Step 1: Since roots are 1, 2, 3, 4: p(x) = (x-1)(x-2)(x-3)(x-4). Step 2: Expand: (x-1)(x-4) = x² - 5x + 4 and (x-2)(x-3) = x² - 5x + 6. Step 3: p(x) = (x² - 5x + 4)(x² - 5x + 6). Let u = x² - 5x: p(x) = (u+4)(u+6) = u² + 10u + 24 = (x²-5x)² + 10(x²-5x) + 24. Step 4: Expand: p(x) = x⁴ - 10x³ + 25x² + 10x² - 50x + 24 = x⁴ - 10x³ + 35x² - 50x + 24. Step 5: Verify p(0) = 24 ✓. Step 6: a = -10, b = 35, c = -50. Therefore a + b + c = -10 + 35 - 50 = -25.

Question 23

If α and β are roots of x² - 5x + 6 = 0, and γ and δ are roots of x² - 7x + 12 = 0, then the value of (α + γ)(β + δ) + (α + δ)(β + γ) is:

Accepted Answer

Step 1: Find roots of first equation: x² - 5x + 6 = 0 → (x-2)(x-3) = 0 → α = 2, β = 3. Step 2: Find roots of second equation: x² - 7x + 12 = 0 → (x-3)(x-4) = 0 → γ = 3, δ = 4. Step 3: Calculate (α + γ)(β + δ) = (2+3)(3+4) = 5×7 = 35. Step 4: Calculate (α + δ)(β + γ) = (2+4)(3+3) = 6×6 = 36. Step 5: Sum = 35 + 36 = 71.

Question 24

If (x + 3) is a factor of the polynomial x² + 7x + 12, what is the other linear factor?

Accepted Answer

Step 1: We need to factor x² + 7x + 12. Step 2: Find two numbers that multiply to 12 and add to 7: these are 3 and 4. Step 3: x² + 7x + 12 = (x + 3)(x + 4). Step 4: Since (x + 3) is one factor, the other factor is (x + 4).

Question 25

What is the sum of the coefficients of the polynomial q(x) = 3x⁴ - 2x³ + 5x - 8?

Accepted Answer

Step 1: The sum of all coefficients of a polynomial p(x) is found by substituting x = 1. Step 2: q(1) = 3(1)⁴ - 2(1)³ + 5(1) - 8 = 3 - 2 + 5 - 8 = -2. Step 3: Therefore, the sum of coefficients is -2.

Question 26

Simplify: (x + 2)(x - 2) - (x + 1)²

Accepted Answer

Step 1: Expand (x + 2)(x - 2) using difference of squares: (x + 2)(x - 2) = x² - 4. Step 2: Expand (x + 1)²: (x + 1)² = x² + 2x + 1. Step 3: Subtract: (x² - 4) - (x² + 2x + 1) = x² - 4 - x² - 2x - 1 = -2x - 5.

Question 27

If (x + 3) is a factor of the polynomial r(x) = x² + 7x + 12, what is the value of r(−3)?

Accepted Answer

Step 1: By the Factor Theorem, if (x + 3) is a factor of r(x), then r(−3) = 0. Step 2: Verify by substitution: r(−3) = (−3)² + 7(−3) + 12 = 9 − 21 + 12 = 0. Therefore, r(−3) = 0.

Question 28

What is the degree of the polynomial q(x) = 7x⁴ − 3x² + 2x + 8?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial q(x) = 7x⁴ − 3x² + 2x + 8. Step 2: The highest power is x⁴, which has exponent 4. Step 3: The degree of a polynomial is the highest exponent of the variable. Therefore, the degree is 4.

Question 29

What is the degree of the polynomial 4x⁵ + 3x³ - 2x + 7?

Accepted Answer

Step 1: The degree of a polynomial is the highest power of x with a non-zero coefficient. Step 2: In 4x⁵ + 3x³ - 2x + 7, the highest power is x⁵. Step 3: Therefore, the degree is 5.

Question 30

If the polynomial p(x) = x³ - 6x² + 11x - 6 has (x - 1) as a factor, what is the remainder when p(x) is divided by (x - 2)?

Accepted Answer

Step 1: By the Remainder Theorem, the remainder when p(x) is divided by (x - a) is p(a). Step 2: We need the remainder when dividing by (x - 2), so we calculate p(2). Step 3: p(2) = (2)³ - 6(2)² + 11(2) - 6 = 8 - 24 + 22 - 6 = 0.

Question 31

Simplify: (x + 2)(x − 2) − (x + 1)²

Accepted Answer

Step 1: Expand (x + 2)(x − 2) using the difference of squares formula: (x + 2)(x − 2) = x² − 4. Step 2: Expand (x + 1)² using the perfect square formula: (x + 1)² = x² + 2x + 1. Step 3: Subtract: (x² − 4) − (x² + 2x + 1) = x² − 4 − x² − 2x − 1 = −2x − 5. Therefore, the answer is −2x − 5.

Question 32

The polynomial s(x) = x³ - 4x² + kx - 8 is divisible by (x - 2). If the quotient is a quadratic polynomial with leading coefficient 1, find the value of k.

Accepted Answer

Step 1: If s(x) is divisible by (x - 2), then s(2) = 0. Step 2: s(2) = (2)³ - 4(2)² + k(2) - 8 = 8 - 16 + 2k - 8 = 2k - 16 = 0. Step 3: 2k = 16 ⟹ k = 8. Step 4: Verify quotient: s(x) = (x - 2)(x² - 2x + 4), which is quadratic with leading coefficient 1. ✓

Question 33

If α and β are the roots of x² - 7x + 12 = 0, find the value of α² + β².

Accepted Answer

Step 1: From Vieta's formulas, α + β = 7 and αβ = 12. Step 2: Use the identity α² + β² = (α + β)² - 2αβ. Step 3: α² + β² = (7)² - 2(12) = 49 - 24 = 25.

Question 34

If (x - 3) is a factor of the polynomial q(x) = x³ - 6x² + 11x - 6, what is the remainder when q(x) is divided by (x - 3)?

Accepted Answer

Step 1: By the Factor Theorem, if (x - 3) is a factor of q(x), then q(3) = 0. Step 2: Verify: q(3) = (3)³ - 6(3)² + 11(3) - 6 = 27 - 54 + 33 - 6 = 0. Step 3: The remainder is 0.

Question 35

If (x - 3) is a factor of the polynomial x³ - 6x² + 11x - 6, find the other two factors.

Accepted Answer

Step 1: Since (x - 3) is a factor, divide x³ - 6x² + 11x - 6 by (x - 3) using synthetic division or long division. Step 2: x³ - 6x² + 11x - 6 = (x - 3)(x² - 3x + 2). Step 3: Factor x² - 3x + 2 = (x - 1)(x - 2). Step 4: The other two factors are (x - 1) and (x - 2).

Question 36

If the polynomial r(x) = x³ + ax² - 5x + 6 leaves a remainder of 4 when divided by (x - 1), find the value of a.

Accepted Answer

Step 1: By the Remainder Theorem, when r(x) is divided by (x - 1), the remainder is r(1). Step 2: r(1) = (1)³ + a(1)² - 5(1) + 6 = 1 + a - 5 + 6 = a + 2. Step 3: Given remainder = 4, so a + 2 = 4 ⟹ a = 2.

Question 37

If p(x) = x⁴ - 3x³ + 2x² + 5x - 1 and q(x) = x² - 2x + 1, find the degree of the quotient when p(x) is divided by q(x).

Accepted Answer

Step 1: When dividing polynomial p(x) by polynomial q(x), the degree of the quotient is degree(p) - degree(q). Step 2: degree(p) = 4 and degree(q) = 2. Step 3: degree(quotient) = 4 - 2 = 2.

Question 38

If p(x) = x³ − 6x² + 11x − 6 is a polynomial, and (x − 1) is a factor, find the sum of the remaining two roots after factoring out (x − 1).

Accepted Answer

Step 1: Since (x − 1) is a factor, we can write p(x) = (x − 1)·q(x) where q(x) is a quadratic. Step 2: Divide p(x) by (x − 1) using synthetic division or long division. Dividing x³ − 6x² + 11x − 6 by (x − 1) gives q(x) = x² − 5x + 6. Step 3: For the quadratic x² − 5x + 6, by Vieta's formulas, the sum of roots = −(coefficient of x)/(coefficient of x²) = −(−5)/1 = 5. Therefore, the sum of the remaining two roots is 5.

Question 39

The polynomial p(x) = x² + bx + 8 has (x - 2) as a factor. Find the value of b.

Accepted Answer

Step 1: If (x - 2) is a factor, then p(2) = 0 by the Factor Theorem. Step 2: p(2) = (2)² + b(2) + 8 = 0. Step 3: 4 + 2b + 8 = 0 ⟹ 2b + 12 = 0 ⟹ 2b = -12 ⟹ b = -6.

Question 40

A polynomial p(x) when divided by (x - 2) leaves remainder 5, and when divided by (x + 1) leaves remainder -4. Find the remainder when p(x) is divided by (x - 2)(x + 1).

Accepted Answer

Step 1: By the Remainder Theorem, p(2) = 5 and p(-1) = -4. Step 2: When p(x) is divided by (x - 2)(x + 1), the remainder must be a polynomial of degree less than 2, so remainder = ax + b. Step 3: p(x) = q(x)(x - 2)(x + 1) + ax + b. Step 4: Substitute x = 2: p(2) = 2a + b = 5. Step 5: Substitute x = -1: p(-1) = -a + b = -4. Step 6: Solve: From equation 2, b = a - 4. Substitute into equation 1: 2a + (a - 4) = 5 → 3a = 9 → a = 3. Step 7: b = 3 - 4 = -1. Step 8: Remainder = 3x - 1.

Question 41

If the polynomial p(x) = x⁴ + ax³ + bx² + cx + d has roots in arithmetic progression with common difference 2, and the sum of roots is 4, then the product of roots is:

Accepted Answer

Step 1: Let roots be (r-3), (r-1), (r+1), (r+3) with common difference 2. Step 2: Sum of roots = (r-3) + (r-1) + (r+1) + (r+3) = 4r = 4, so r = 1. Step 3: Roots are -2, 0, 2, 4. Step 4: Product of roots = (-2) × 0 × 2 × 4 = 0.

Question 42

If p(x) = x³ - 12x² + 47x - 60 and one root is 3, then the sum of the other two roots is:

Accepted Answer

Step 1: By Vieta's formula, sum of all three roots = 12 (negative of coefficient of x² divided by coefficient of x³). Step 2: If one root is 3, then sum of other two roots = 12 - 3 = 9.

Question 43

A polynomial p(x) of degree 3 satisfies p(0) = 2, p(1) = 3, p(2) = 12, and p(3) = 35. The coefficient of x² in p(x) is:

Accepted Answer

Step 1: Let p(x) = ax³ + bx² + cx + d. Step 2: From p(0) = 2, we get d = 2. Step 3: From p(1) = 3, we get a + b + c + 2 = 3, so a + b + c = 1. Step 4: From p(2) = 12, we get 8a + 4b + 2c + 2 = 12, so 8a + 4b + 2c = 10, or 4a + 2b + c = 5. Step 5: From p(3) = 35, we get 27a + 9b + 3c + 2 = 35, so 27a + 9b + 3c = 33, or 9a + 3b + c = 11. Step 6: Solve the system: a + b + c = 1, 4a + 2b + c = 5, 9a + 3b + c = 11. Subtracting eq1 from eq2: 3a + b = 4. Subtracting eq2 from eq3: 5a + b = 6. Subtracting: 2a = 2, so a = 1. Then b = 4 - 3(1) = 1. Then c = 1 - 1 - 1 = -1. Step 7: The coefficient of x² is b = 1.

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Polynomials — Practice Questions

60-Second Revision — Polynomials