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

If $x^{3} + a x^{2} + b x + 6$ has $(x - 2)$ as a factor and leaves a remainder $3$ when divided by $(x - 3)$ , find the values of $a$ and $b$ .

Accepted Answer

Let, $f (x) = x^{3} + a x^{2} + b x + 6$

By using remainder theorem, if we divide $f (x)$ by $(x - 3)$ then it will leave a remainder as $f (3)$ .

So, $f (3) = 3$

$\Rightarrow 3^{3} + a \times 3^{2} + b \times 3 + 6 = 3$

$\Rightarrow 27 + 9 a + 3 b + 6 = 3$

$\Rightarrow 9 a + 3 b + 33 = 3$

$\Rightarrow 9 a + 3 b = 3 - 33$

$\Rightarrow 9 a + 3 b = - 30$

$\Rightarrow 3 a + b = - 10$ _____________ $(1)$

It is also given that, $(x - 2)$ is a factor of $f (x)$ .

By using factor theorem, we get, $(x - a)$ is the factor of $f (x)$ , if $f (a) = 0$ and also $f (2) = 0$ .

Now, $f (2) = 0$

$\Rightarrow 2^{3} + (a \times 2^{2}) + (b \times 2) + 6 = 0$

$\Rightarrow 8 + 4 a + 2 b + 6 = 0$

$\Rightarrow 4 a + 2 b = - 14$

$\Rightarrow 2 a + b = - 7$ _______________ $(2)$

Now by subtracting $(2)$ from $(1)$ , we get,

$\Rightarrow a = - 3$

Putting the value of $a$ in $(1)$ , we get

$3 (- 3) + b = - 10$

$\Rightarrow - 9 + b = - 10$

$\Rightarrow b = - 10 + 9$

$\Rightarrow b = - 1$

Therefore, $b = - 1$ and $b = - 3$

If $x^{3} + a x^{2} + b x + 6$ has $(x - 2)$ as a factor and leaves a remainder $3$ when divided by $(x - 3)$ , find the values of $a$ and $b$ .

Important Questions on Polynomials

Learn and Prepare for any exam you want !

If x3+ax2+bx+6 has x-2 as a factor and leaves a remainder 3 when divided by x-3, find the values of a and b.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

If $x^{3} + a x^{2} + b x + 6$ has $(x - 2)$ as a factor and leaves a remainder $3$ when divided by $(x - 3)$ , find the values of $a$ and $b$ .