Polynomials (Class 10 Mathematics)

Polynomials are algebraic expressions with terms containing variables raised to non-negative integer powers. Understanding zeros, factors, and the relation

Polynomials (Class 10 Mathematics)

Polynomials are algebraic expressions with terms containing variables raised to non-negative integer powers. Understanding zeros, factors, and the relation

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TL;DR: Polynomials are algebraic expressions with terms containing variables raised to non-negative integer powers. Understanding zeros, factors, and the rel…

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Polynomials are algebraic expressions with terms containing variables raised to non-negative integer powers. Understanding zeros, factors, and the relation

Polynomials MCQs with Answers & Explanations

Q1. A polynomial of degree 2 is called:

  1. Linear
  2. Quartic
  3. Cubic
  4. Quadratic ✓ (correct)

Explanation: A quadratic polynomial has degree 2, with the general form ax² + bx + c where a ≠ 0.

Q2. If (x - 2) is a factor of polynomial p(x), then p(2) equals:

  1. 0 ✓ (correct)
  2. -1
  3. 1
  4. 2

Explanation: By the Factor Theorem, if (x - a) is a factor of p(x), then p(a) = 0.

Q3. The zeros of the polynomial x² - 5x + 6 are:

  1. 5 and 6
  2. 2 and 3 ✓ (correct)
  3. -2 and -3
  4. 1 and 6

Explanation: x² - 5x + 6 = (x - 2)(x - 3), so the zeros are x = 2 and x = 3.

Q4. If α and β are zeros of ax² + bx + c, then α + β equals:

  1. b/a
  2. -c/a
  3. -b/a ✓ (correct)
  4. c/a

Explanation: By Vieta's formulas for a quadratic, the sum of zeros = -b/a.

Q5. The remainder when p(x) = x³ + 2x² - x + 1 is divided by (x - 1) is:

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

Explanation: By the Remainder Theorem, the remainder = p(1) = 1 + 2 - 1 + 1 = 3.

Q6. A cubic polynomial has at most ____ zeros.

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

Explanation: A polynomial of degree n has at most n real zeros. A cubic (degree 3) polynomial has at most 3 zeros.

Q7. If the zeros of a polynomial are 1, 2, and 3, the polynomial is:

  1. (x-1)² + (x-2)² + (x-3)²
  2. (x-1)(x-2)(x-3) ✓ (correct)
  3. x³ - x² - x + 3
  4. (x+1)(x+2)(x+3)

Explanation: If the zeros are 1, 2, and 3, then the polynomial with leading coefficient 1 is (x-1)(x-2)(x-3).

Q8. The product of zeros of ax² + bx + c equals:

  1. b/c
  2. -c/a
  3. c/a ✓ (correct)
  4. -b/a

Explanation: By Vieta's formulas, the product of zeros = c/a for a quadratic polynomial.

Q9. Which polynomial has a zero at x = -1?

  1. x² + x + 1
  2. x² + 1
  3. x² - x + 1
  4. x² - 1 ✓ (correct)

Explanation: If x = -1: (-1)² - 1 = 1 - 1 = 0. So x = -1 is a zero of x² - 1.

Q10. The degree of the polynomial (x+1)³(x-2)²(x+3) is:

  1. 6 ✓ (correct)
  2. 3
  3. 4
  4. 5

Explanation: Degree = 3 (from (x+1)³) + 2 (from (x-2)²) + 1 (from (x+3)) = 6.

Q11. If x = 2 is a zero of the polynomial p(x) = x^2 - 2x + k, then k is:

  1. 2
  2. 0 ✓ (correct)
  3. -2
  4. 4

Explanation: If x = 2 is a zero, then p(2) = 0. So 4 - 4 + k = 0, which gives k = 0.

Q12. The degree of the polynomial 4x^3 - 5x^2 + 2x - 1 is:

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

Explanation: The degree of a polynomial is the highest power of the variable, which is 3 in this case.

Q13. If the sum of zeros of the polynomial x^2 - 5x + 6 is s, then s is:

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

Explanation: For a quadratic ax^2 + bx + c, sum of zeros = -b/a = -(-5)/1 = 5.

Q14. The product of zeros of the polynomial 2x^2 - 8x + 6 is:

  1. 3 ✓ (correct)
  2. -4
  3. 4
  4. -3

Explanation: For a quadratic ax^2 + bx + c, product of zeros = c/a = 6/2 = 3.

Q15. When p(x) = x^3 - 2x^2 + x - 1 is divided by (x - 1), the remainder is:

  1. 0
  2. -1 ✓ (correct)
  3. -2
  4. -3

Explanation: By the Remainder Theorem the remainder is p(1) = 1 - 2 + 1 - 1 = -1.

Q16. Which of the following is a factor of x^3 - 8?

  1. x + 2
  2. x - 2
  3. x^2 + 2x + 4
  4. All of the above ✓ (correct)

Explanation: x^3 - 8 = (x - 2)(x^2 + 2x + 4), so all three are factors of this polynomial.

Q17. The number of zeros of a cubic polynomial is at most:

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

Explanation: A polynomial of degree n has at most n zeros. A cubic polynomial has degree 3, so it has at most 3 zeros.

Q18. If (x - 1) is a factor of x^3 + ax^2 - 2x - 1, then a is:

  1. 0
  2. -1
  3. 1
  4. 2 ✓ (correct)

Explanation: If (x - 1) is a factor, then p(1) = 0. So 1 + a - 2 - 1 = 0, which gives a = 2.

Q19. The zeros of the polynomial (x - 3)(x + 2)(x - 1) are:

  1. 3, -2, 1 ✓ (correct)
  2. -3, 2, -1
  3. -3, -2, -1
  4. 3, 2, 1

Explanation: The zeros are the values of x that make the polynomial equal to 0, which are 3, -2, and 1.

Q20. If the polynomial x^2 - px + q has zeros a and b, then:

  1. a + b = q, ab = p
  2. a + b = p, ab = q ✓ (correct)
  3. a + b = -p, ab = -q
  4. a + b = p, ab = -q

Explanation: By Vieta's formulas, sum of zeros = p and product of zeros = q for the polynomial x^2 - px + q.

Frequently Asked Questions

What is the Factor Theorem and how is it useful?

The Factor Theorem states that (x - a) is a factor of p(x) if and only if p(a) = 0. It's useful for finding zeros and factoring polynomials without lengthy division.

How can Vieta's formulas help solve problems?

Vieta's formulas relate the coefficients of a polynomial to the sum and product of its zeros. For a quadratic ax² + bx + c with zeros α and β: α + β = -b/a and αβ = c/a. This helps find zeros or construct polynomials from given zeros.

More Class 10 Mathematics MCQs

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