How do you factor 4x^3+4x^2-7x+2?

Answer 1

You first look at the magnitudes of the terms when x is 1. They are 4, 4, 7 and 2. There is no way we can juggle their signs to make them add to zero.

Next you look at the magnitudes when x is 2. They are 32, 16, 14, and 2. Yippee! If one of the choices of sign can give us -32 + 16 + 14 +2, we have a root. Yes, x = -2 makes the terms add to zero, so (x + 2) is a factor.

By synthetic division, 4x^3 + 4x^2 - 7x + 2 = (x + 2) (4x^2 - 4x + 1), and the quadratic factors as (2x - 1)^2. I don't know why everybody else got it wrong.

Answer 2

The rational root theorem says that if a polynomial has any rational roots, they are factors of the lowest-order coefficient (2) divided by factors of the highest-order coefficient (4). Try x=1/2, and sure enough, it is a root: 4(1/2)^3 + 4(1/2)^2 - 7(1/2) + 2 = 0.

Divide this polynomial by (x-1/2). Divide leading terms: 4x^3 / x = 4x^2. Then what's left over is
4x^3 + 4x^2 - 7x + 2 - 4x^2(x-1/2) =
4x^3 + 4x^2 - 7x + 2 - 4x^3 + 2x^2 =
6x^2 - 7x + 2.

Now divide this by (x-1/2). Dividing leading terms gives 6x^2 / x = 6x, and
6x^2 - 7x + 2 - 6x(x-1/2) =
6x^2 - 7x + 2 - 6x^2 + 3x =
-4x + 2 = -4(x-1/2), so the quotient is 4x^2 + 6x - 4.

To factor this, first factor out a 2: 2(2x^2 + 3x - 2).
This factors into 2(2x - 1)(x + 2), as can be determined by trial and error: looking for a factorization of the form (ax+b)(cx+d), where ab=2, bc+ad=3, and bd=-2. Try a=2, c=1, so now you're looking for (2x+b)(x+d) where b+2d=3 and bd=-2. One of b and d must be negative, and the other positive, and since b+2d is positive, b is probably the negative one. Trying b=-1 and d=2 gives the factorization.

Answer 3

4x^3 +4x^2 -7x+2

rewrite -7x as x - 8x
4x^3 +4x^2 + x - 8x +2
(4x^3 +4x^2 + x) - (8x +2)
factor out x in (4x^3 +4x^2 + x) and 2 in (8x +2)
x(4x^2 +4x + 1) - 2 (4x+1)
x(2x+1)^2 - 2 (4x+1)

oh oh

i dont see any common term
im stuck

Answer 4

TO CONTINUE THIS ANSWER:x(2x+1)^2 - 2 (4x+1)

Set each term equal to zero
x=0; 2x+1=0; 2x+1=0; 2(4x+1)=0
x=0; 2x=-1; 4x+1=0
x=0; x= (-1/2) 4x=-1
x=(-1/4)
so x= 0, (-1/2), (-1/2), (-1/4)

Answer 5

im not sure but 4x^2(x+1)-7x+2