Solve by Factoring 4x^3-8x^2-x+2=0

$4 x^{3} - 8 x^{2} - x + 2 = 0$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{3} - 8 x^{2}) - x + 2 = 0$

Factor out the greatest common factor (GCF) from each group.

$4 x^{2} (x - 2) - (x - 2) = 0$

Step 2

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$(x - 2) (4 x^{2} - 1) = 0$

Step 3

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$(x - 2) ({(2 x)}^{2} - 1) = 0$

Step 4

Rewrite $1$ as $1^{2}$ .

$(x - 2) ({(2 x)}^{2} - 1^{2}) = 0$

Step 5

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$(x - 2) ((2 x + 1) (2 x - 1)) = 0$

Remove unnecessary parentheses.

$(x - 2) (2 x + 1) (2 x - 1) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$2 x + 1 = 0$

$2 x - 1 = 0$

Step 7

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 8

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 9

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 10

The final solution is all the values that make $(x - 2) (2 x + 1) (2 x - 1) = 0$ true.

$x = 2, - \frac{1}{2}, \frac{1}{2}$

Solve by Factoring 4x^3-8x^2-x+2=0

Related Questions

Support LCMGCF.com by disabling your adblocker.