Name: Factor Theorem: Illustration
Uploaded: 2019-07-19
Duration: 4 min 17 s
Description: The factorisation of a cubic polynomial might look difficult. Let us learn how to find factors of a cubic polynomial with an illustration. Watch the video fo...

Question

In the pair of polynomials $x^{2} - 3 x + 5, x - 4$ , find the number to be subtracted from the first to get a polynomial for which the second is a factor. Find also the second factor of the polynomial got on subtracting the number.

Accepted Answer

Given: $p (x) = x^{2} - 3 x + 5$ and $x - 4$ .

We need to find the number to be subtracted form the first to get a polynomial for which the second is a factor.

For any second degree polynomial $p (x)$ and for any number $a$ , the polynomial $x - a$ is a factor of the polynomial $p (x) - p (a)$

Then, the polynomial $x - 4$ is a factor of the polynomial $p (x) - p (4)$

Now, $p (4) = 4^{2} - 3 \times 4 + 5 = 9$

So, we need to subtract $9$ from the first to get a polynomial for which the second is a factor.

That is, $p (x) - p (4) = x^{2} - 3 x + 5 - 9$

$\Rightarrow p (x) - p (4) = x^{2} - 3 x - 4$

We need to find the second factor of the polynomial $x^{2} - 3 x - 4$ .

Now, express $x^{2} - 3 x - 4 = (x - a) (x - b)$ form.

So, $x^{2} - 3 x - 4 = x^{2} - (a + b) x + a b$

By comparing $LHS$ and $RHS$ of the above equation, we get $a + b = 3$ and $a b = - 4$

We must find the two numbers with sum $3$ and product $- 4$ .

The two numbers are $4$ and $- 1$ , thus we can take $a = 4$ and $b = - 1$ and we can write $x^{2} - 3 x - 4 = (x - 4) (x + 1)$

Hence, the second factor is $x + 1$ .

Kerala Board Solutions for Chapter: Polynomials, Exercise 2: Exercise 2

Kerala Board Mathematics Solutions for Exercise - Kerala Board Solutions for Chapter: Polynomials, Exercise 2: Exercise 2

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