Math question from Algebra 2

Question

Write h(x) = -3x⁴ +21x³ - 48x² +36x in terms of its linear factors if (x-3) is one of the factors.

Answer 1

–3x is a common factor in all terms so

h(x) = -3x⁴ +21x³ - 48x² + 36x = –3x(x³ – 7x² + 16x – 12)

We’re given that (x – 3) is a factor, so

(x³ – 7x² + 16x – 12)/(x – 3) = x² – 4x + 4 is found by long division

x² – 4x + 4 = (x – 2)²

So h(x) = -3x⁴ +21x³ - 48x² + 36x = –3x(x – 3)(x – 2)²

Answer 2

This function has a common factor of -3x. Removing this factor reduces the power.

So, -3x⁴ + 12x³ - 48x² + 36X = -3x ( x³ - 7x² + 16x - 12)

We are told that x - 3 is also a factor. So division can reduce the power again.

This can be done by synthetic division or by division of polynomials. I choose the latter.

X² - 4x + 4_______

X - 3 ) x³ - 7 x² + 16 x - 12

x³ - 3 x²

-4 x^2.+ 16x

-4 x² + 12x

4 x - 12

4 x - 12

0

Now we have -3x⁴ + 12x³ - 48x² + 36X = -3x ( x³ - 7x² + 16x - 12) =. -3x ( x - 3) ( X² - 4x + 4)

Now the last trinomial can be factored into ( x - 2 )(x - 2)

So the final result is -3x ( x - 3) (x - 2) ( x - 2) OR. -3x ( x - 3) ( x - 2)²

^

Answer 3

Factor x out so that one root is 0.

Then divide the reduced h(x) by x-3 using long division.

Then factor the quotient to obtain the other 2 roots.