step by step factor:

Question

32x⁴-72x³+54x

Answer 1

We can first simplify this by dividing by 2 and discarding it and then factoring out x.

x(16x³-36x²+27)

Let's just concentrate on the polynomial in the parenthesis using the Rational Root Theorem.

That means we want to look at all rational root of a₀/a_n.

a₀ for this polynomial is 27 which has factors ±{1,3,9}.

a_n for this polynomial is 16 which has factors ±{1,2,4,8,16}

So possible roots are ±{1/1,1/2,1/4,1/8,1/16, 3/2, 3/4, 3/8, 3/16, 9/1,9/2,9/4,9/8,9/16}

Substituting each of for x in 16x³-36x²+27 we find that only 3/2 and -3/4 will produce a value of 0. With a₀ being 27, picking the ones with 3 in the numerator first helps speed up the selection.

So we can factor out (2x-3) and (4x+3).

You can use synthetic division to divide out (4x+3).

16 -36 0 27 |-3/4

-12 36 -27

------------------------

16 -48 36 0

16 -48 36 |3/2

24 -36

-----------------

16 -24 0 --> 16x-24 --> x = 24/16 = 3/2 --> (2x-3) again

So the factors are x(2x-3)²(4x+3) or x = 0,3/2, 3/2, -3/4