Polynomials: Zeros MCQs Quiz | Class 10

This quiz on Polynomials: Zeros MCQs is designed for Class X Mathematics (Code 041), part of Unit II: Algebra. It specifically focuses on finding zeros of polynomials, with a strong emphasis on quadratic polynomials. Test your understanding, then submit to see your score and download a detailed answer PDF for review.

Understanding Zeros of Polynomials

The zeros of a polynomial are the values of the variable for which the polynomial evaluates to zero. Graphically, these are the x-intercepts, where the graph of the polynomial crosses or touches the x-axis. Understanding zeros is fundamental in algebra, especially for solving equations and analyzing polynomial behavior.

What are Zeros?

For a polynomial P(x), a real number ‘a’ is a zero if P(a) = 0. For example, if P(x) = x – 2, then P(2) = 2 – 2 = 0, so 2 is a zero of P(x).

Zeros of Quadratic Polynomials (ax2 + bx + c)

A quadratic polynomial, P(x) = ax2 + bx + c (where a is not equal to 0), has at most two zeros. These can be found using several methods:

  1. Factoring: If the quadratic can be factored into linear terms, set each factor to zero to find the roots. For example, for x2 – 5x + 6 = (x – 2)(x – 3), the zeros are x=2 and x=3.
  2. Quadratic Formula: For any quadratic polynomial, the zeros are given by:

    x = [-b ± sqrt(b2 – 4ac)] / 2a

    This formula is universally applicable for finding real or complex zeros. The term (b2 – 4ac) is called the discriminant, which determines the nature of the zeros.
    • If b2 – 4ac > 0, there are two distinct real zeros.
    • If b2 – 4ac = 0, there are two equal real zeros (or one repeated zero).
    • If b2 – 4ac < 0, there are no real zeros (two complex conjugate zeros).

Relationship between Zeros and Coefficients

For a quadratic polynomial P(x) = ax2 + bx + c, if alpha (α) and beta (β) are its zeros, then there’s a direct relationship with its coefficients:

Relationship Formula
Sum of Zeros (α + β) -b/a
Product of Zeros (αβ) c/a

This relationship is extremely useful for forming a quadratic polynomial given its zeros or for finding the sum/product of zeros without actually calculating them.

Graphical Interpretation

The zeros of a polynomial P(x) are the x-coordinates of the points where the graph of y = P(x) intersects the x-axis. A quadratic polynomial’s graph is a parabola. It can intersect the x-axis at two distinct points, one point (if the parabola touches the x-axis at its vertex), or not at all (if the parabola is entirely above or below the x-axis).

Quick Revision Points

  • A zero of P(x) is a value ‘a’ such that P(a) = 0.
  • A polynomial of degree ‘n’ has at most ‘n’ zeros.
  • For a quadratic P(x) = ax2 + bx + c, zeros can be found by factoring or using the quadratic formula.
  • Sum of zeros = -b/a, Product of zeros = c/a for a quadratic polynomial.
  • Graphically, zeros are the x-intercepts.

Practice Questions

  1. Find the zeros of the polynomial x2 – 7x + 10.
  2. Determine a quadratic polynomial whose zeros are -2 and 5.
  3. If the sum of the zeros of a quadratic polynomial is 3 and the product is -10, write the polynomial.
  4. For the polynomial 2x2 – 8x + 6, find the sum and product of its zeros.
  5. What is the value of ‘k’ if -1 is a zero of the polynomial x2 – (k+1)x + 2?