Name: Factorisation Using Difference of Two Squares
Uploaded: 2019-07-19
Duration: 5 min 14 s
Description: Sometimes writing a polynomial as the difference of two squares is the only way to factorise it. So let’s learn how to do it in this simple video!

Question

By actual division, show that $(x + 2)$ is a factor of the polynomial $p (x) = 2 x^{4} + x^{3} - 14 x^{2} - 19 x - 6$ . Verify it using factor theorem.

Accepted Answer

We have $p (x) = 2 x^{4} + x^{3} - 14 x^{2} - 19 x - 6$ and $g (x) = (x + 2)$ .

Perform $p (x) \div g (x)$ as follows.

$\begin{matrix} 2 x^{3} & - & 3 x^{2} & - & 8 x & - & 3 \\ x + 2 & 2 x^{4} & + & x^{3} & - & 14 x^{2} & - & 19 x & - & 6 \\ 2 x^{4} & + & 4 x^{3} \\ - 3 x^{3} & - & 14 x^{2} & - & 19 x & - & 6 \\ - 3 x^{3} & - & 6 x^{2} \\ - 8 x^{2} & - & 19 x & - & 6 \\ - 8 x^{2} & - & 16 x \\ - 3 x & - & 6 \\ - 3 x & - & 6 \\ 0 \end{matrix}$

Since, in $p (x) \div g (x)$ , the remainder is zero, we have $(x + 2)$ is a factor of $2 x^{4} + x^{3} - 14 x^{2} - 19 x - 6$ .

Factor Theorem: If $p (x)$ is a polynomial of degree $\geq 1$ and $α$ is real number, then $x - α$ is a factor of $p (x)$ if $p (α) = 0$ and conversely, $p (α) = 0$ if $x - α$ is a factor of $p (x)$ .

Now, $p (- 2) = 2 {(- 2)}^{4} + {(- 2)}^{3} - 14 {(- 2)}^{2} - 19 (- 2) - 6$

$\Rightarrow p (- 2) = 32 - 8 - 56 + 38 - 6$

$\Rightarrow p (- 2) = 70 - 70 = 0$

$\Rightarrow p (- 2) = 0$

$\Rightarrow (x + 2)$ is a factor of $2 x^{4} + x^{3} - 14 x^{2} - 19 x - 6$

Thus, we showed that $(x + 2)$ is a factor of $p (x) = 2 x^{4} + x^{3} - 14 x^{2} - 19 x - 6$ by actual division and verified it using factor theorem.

By actual division, show that $(x + 2)$ is a factor of the polynomial $p (x) = 2 x^{4} + x^{3} - 14 x^{2} - 19 x - 6$ . Verify it using factor theorem.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

By actual division, show that x+2 is a factor of the polynomial px=2x4+x3-14x2-19x-6. Verify it using factor theorem.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

By actual division, show that $(x + 2)$ is a factor of the polynomial $p (x) = 2 x^{4} + x^{3} - 14 x^{2} - 19 x - 6$ . Verify it using factor theorem.