NCERT Class 9 Solutions: Polynomials (Chapter 2) Exercise 2.4 – Part 1

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Q-1 Determine which of the following polynomials has Equation a factor:

  1. Equation

  2. Equation

  3. Equation

  4. Equation

Solution:

Understanding relation between polynomial and its roots

Polynomials and Roots

Understanding relation between polynomial and its roots

If Equation is a factor of a polynomial Equation , then we can write, Equation Since zero multiplied by any number is zero. Therefore, this polynomial Equation would become zero for all the values of x where either Equation becomes zero or Equation becomes zero. We don’t know anything about g(x) but we know that Equation becomes zero for Equation . Therefore, Equation Therefore, if x + 1 is a factor of p(x), p(x) must become zero at x = -1. For each polynomial above we would test its value at x = -1. If it comes out to be zero (x+1) is a factor of that polynomial.

  1. Let Equation

    Equation

    Equation

    Equation

    Equation Is a factor of polynomial Equation

  2. Let Equation

    Equation

    Equation

    Equation

    Equation Is not a factor of polynomial Equation

  3. Let Equation

    Equation

    Equation

    Equation

    Equation

    Equation Is not a factor of polynomial Equation

  4. Let Equation

    Equation

    Equation

    Equation

    Equation Is not a factor of polynomial Equation

Q-2 Use the Factor Theorem to determine whether Equation is a factor of Equation in each of the following cases:

  1. Equation

  2. Equation

  3. Equation

Solution:

  1. Equation

    Zero of Equation is Equation .

    If Equation is a factor of Equation then p must also have a zero at -1, i.e. Equation

    Equation

    Equation

    Equation

    Equation Is a factor of polynomial Equation .

  2. Equation

    Zero of Equation is Equation .

    If Equation is a factor of Equation then p must also have a zero at -2, i.e. Equation

    Equation

    Equation

    Equation

    Equation Is not a factor of polynomial Equation .

  3. Equation

    Zero of Equation is Equation .

    If Equation is a factor of Equation then p must also have a zero at 3, i.e. Equation

    Equation

    Equation

    Equation

    Equation Is a factor of polynomial Equation .

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