Class 9 chapter polynomials.
Factorise x^4 - 7x^3 + 9x^2 + 7x - 10 given that x - 1 is the factor.
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R.D. Sharma Solutions»R.D. Sharma Class 9 Solutions»Chapter 6 Factorization of Polynomials»Factorization of Polynomials Exercise 6.5
Chapter 6: Factorization of Polynomials Exercise – 6.5
Question: 1
Using factor theorem, factorize of the polynomials:
x3 + 6x2 + 11x + 6
Solution:
Given polynomial, f(x) = x3 + 6x2 + 11x + 6
The constant term in f(x) is 6
The factors of 6 are ± 1, ± 2, ± 3, ± 6
Let, x + 1 = 0
=> x = -1
Substitute the value of x in f(x)
f(-1) = (−1)3 + 6(−1)2 + 11(−1) + 6
= - 1 + 6 - 11 + 6
= 12 – 12
= 0
So, (x + 1) is the factor of f(x)
Similarly, (x + 2) and (x + 3) are also the factors of f(x)
Since, f(x) is a polynomial having a degree 3, it cannot have more than three linear factors.
∴ f(x) = k(x + 1)(x + 2)(x + 3)
=> x3 + 6x2 + 11x + 6 = k(x + 1)(x + 2)(x + 3)
Substitute x = 0 on both the sides
=> 0 + 0 + 0 + 6 = k(0 +1)(0 + 2)(0 + 3)
=> 6 = k(1*2*3)
=> 6 = 6k
=> k = 1
Substitute k value in f(x) = k(x + 1)(x + 2)(x + 3)
=> f(x) = (1)(x + 1)(x + 2)(x + 3)
=> f(x) = (x + 1)(x + 2)(x + 3)
∴ x3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3)
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