Question

The polynomial p(x)=x^3+3x^2-4 has a known factor of (x-1). Rewrite p(x) as a product of linear factors

Answer 1

`p(x) = x^(3) + 3x^(2)   -4 = (x-1)(x+2)(x+2)`

Explanation:

The given polynomial
`p(x) = x^(3) + 3x^(2)   -4`

Now, given that (x-1) is a factor of the above equation.

Now, divide the given polynomial with the factor (x-1)

By Long division, we get

Quotient =
`4x^(2) + 4x + 4` and Remainder = 0

So, by the Remainder theorem

`p(x) = x^(3) + 3x^(2)   -4 = (4x^(2) + 4x + 4) * (x-1)`

Now, Simplifying the quotient further, we get

`4x^(2) + 4x + 4` =
`4x^(2) + 2x + 2x+ 4`

=
`x(x+2)+ 2(x+2)`

or,
`4x^(2) + 4x + 4` = (x+2)(x+2)

Hence, the given polynomial
`p(x) = x^(3) + 3x^(2)   -4` can be written as a product of linear factors.

`p(x) = x^(3) + 3x^(2)   -4  = (x-1)(x+2)(x+2)`