Question

find the value of a if (x-a) is a factor of
${x}^{5} - {a}^{2} {x}^{3} + 2x + a + 3$
hence factorise
${x}^{2} - 2ax - 3$

Answer 1

Hello,.

Let
$p(x) = {x}^{5} - {a}^{2} {x}^{3} + 2x + a + 3$
Given that (x-a) is a factor of p(x).

So,
By the factor theorem
X-a =0
Or, x = a

Now putting x=a in p(x)
So,
Again by factor theorem
...

P(a) =0
So,
${x}^{5} - {a}^{2} {x}^{3} + 2x + a + 3 \\ = {a}^{5} - {a}^{2} {a}^{3 } + 2a + a + 3 \\ {a}^{5} - {a}^{5} + 3a + 3 = 0 \\ 3a + 3 = 0 \\ 3a = ( - 3) \\ a = \frac{( - 3)}{3} = ( - 1)$
We got the value of a as (-1)

Now ,

Putting this in other polynomial...
x² - 2(-1)x - 3
Or
x² + 2x - 3

By middle term splitting, we are factorising the polynomial
So,
x² +3x - x - 3
Or, x(x+3) - 1(x+3)
Or, (x+3)(x-1)


Hope this will be helping you ✌️