Write polynomial P as a product of linear factors : P(x) = 5x^3 -2x^2 + 5x - 2

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justaguide eNotes educator| Certified Educator

We have to write the polynomial P(x) = 5x^3 -2x^2 + 5x - 2 as a product of linear factors.

P(x) = 5x^3 -2x^2 + 5x - 2

=> P(x) = x^2( 5x - 2) + 1( 5x - 2)

=> P(x) = (x^2 + 1)(5x - 2)

The term x^2 + 1 cannot be factorized into linear terms.

If instead we had P(x) = 5x^3 -2x^2 - 5x + 2 we could have had P(x) = (5x - 2)(x - 1)(x + 1), which are linear terms.

The factors of P(x) = 5x^3 -2x^2 + 5x - 2 are

(x^2 + 1)(5x - 2).

hala718 eNotes educator| Certified Educator

Given the polynomial P(x) = 5x^3 - 2x^2 + 5x -2

We need to factor P(x).

First we will rearrange the terms of P(x).

==> P(x) = 5x^3 + 5x - 2x^2 - 2

Now we will factor 5x from the first two terms.

==> P(x) = 5x (x^2 +1) - 2x^2 -2

Now we will factor -2 from the last two terms.

==> P(x) = 5x(x^2 +1) -2 ( x^2 + 1)

Now we will factor (x^2+1).

==> P(x) = (x^2 + 1) (5x-2)

Now we will factor (x^2 +1)

==> P(x) = (x-i)(x+i) ( 5x-2)