Question

14. Factor x4 + 3x2 - 28.(x2 - 7)(x - 2)(x + 2)(x2 - 2)(x2 + 14)(x2 + 7)(x - 2)(x + 2)(x2 + 4)(x2 - 7)

Answer 1

`x^4+3x^2-28=(x^2+7)(x-2)(x+2)`

Explanation:

To factorize the expression, we can use a variable substitution. Let's say that z=x^2.

`\begin{gathered} x^4+3x^2-28 \\ z^2+3z-28 \end{gathered}`

Then, to factorize this we need to factor in the form:

`(z+\text{?)(z}+\text{?)}`

The numbers that go in the blanks, have to:

*Add together to get 3

`-4+7=3`

*Multiply together to get -28

`-4\cdot7=-28`

So, we get:

`z^2+3z-28=(z-4)(z+7)`

Substitute the equation z=x^2

`(x^2-4)(x^2+7)`

Factorizing the perfect square binomial:

`x^4+3x^2-28=(x^2+7)(x-2)(x+2)`