Find the Roots (Zeros) V(x)=2x^4-7x^3+x^2+7x-3

$V (x) = 2 x^{4} - 7 x^{3} + x^{2} + 7 x - 3$

Step 1

Set $2 x^{4} - 7 x^{3} + x^{2} + 7 x - 3$ equal to $0$ .

$2 x^{4} - 7 x^{3} + x^{2} + 7 x - 3 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Regroup terms.

$- 7 x^{3} + 7 x + 2 x^{4} + x^{2} - 3 = 0$

Factor $- 7 x$ out of $- 7 x^{3} + 7 x$ .

Factor $- 7 x$ out of $- 7 x^{3}$ .

$- 7 x \cdot x^{2} + 7 x + 2 x^{4} + x^{2} - 3 = 0$

Factor $- 7 x$ out of $7 x$ .

$- 7 x \cdot x^{2} - 7 x \cdot - 1 + 2 x^{4} + x^{2} - 3 = 0$

Factor $- 7 x$ out of $- 7 x (x^{2}) - 7 x (- 1)$ .

$- 7 x (x^{2} - 1) + 2 x^{4} + x^{2} - 3 = 0$

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

$- 7 x (x^{2} - 1^{2}) + 2 x^{4} + x^{2} - 3 = 0$

Factor.

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

$- 7 x ((x + 1) (x - 1)) + 2 x^{4} + x^{2} - 3 = 0$

Remove unnecessary parentheses.

$- 7 x (x + 1) (x - 1) + 2 x^{4} + x^{2} - 3 = 0$

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

$- 7 x (x + 1) (x - 1) + 2 {(x^{2})}^{2} + x^{2} - 3 = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 7 x (x + 1) (x - 1) + 2 u^{2} + u - 3 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

$- 7 x (x + 1) (x - 1) + 2 u^{2} + 1 u - 3 = 0$

Rewrite $1$ as $- 2$ plus $3$

$- 7 x (x + 1) (x - 1) + 2 u^{2} + (- 2 + 3) u - 3 = 0$

Apply the distributive property.

$- 7 x (x + 1) (x - 1) + 2 u^{2} - 2 u + 3 u - 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- 7 x (x + 1) (x - 1) + 2 u^{2} - 2 u + 3 u - 3 = 0$

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

$- 7 x (x + 1) (x - 1) + 2 u (u - 1) + 3 (u - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $u - 1$ .

$- 7 x (x + 1) (x - 1) + (u - 1) (2 u + 3) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$- 7 x (x + 1) (x - 1) + (x^{2} - 1) (2 x^{2} + 3) = 0$

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

$- 7 x (x + 1) (x - 1) + (x^{2} - 1^{2}) (2 x^{2} + 3) = 0$

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

$- 7 x (x + 1) (x - 1) + (x + 1) (x - 1) (2 x^{2} + 3) = 0$

Factor $(x + 1) (x - 1)$ out of $- 7 x (x + 1) (x - 1) + (x + 1) (x - 1) (2 x^{2} + 3)$ .

Factor $(x + 1) (x - 1)$ out of $- 7 x (x + 1) (x - 1)$ .

$(x + 1) (x - 1) (- 7 x) + (x + 1) (x - 1) (2 x^{2} + 3) = 0$

Factor $(x + 1) (x - 1)$ out of $(x + 1) (x - 1) (- 7 x) + (x + 1) (x - 1) (2 x^{2} + 3)$ .

$(x + 1) (x - 1) (- 7 x + 2 x^{2} + 3) = 0$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x + 1) (x - 1) (- 7 u + 2 u^{2} + 3) = 0$

Factor by grouping.

Reorder terms.

$(x + 1) (x - 1) (2 u^{2} - 7 u + 3) = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 u$ .

$(x + 1) (x - 1) (2 u^{2} - 7 u + 3) = 0$

Rewrite $- 7$ as $- 1$ plus $- 6$

$(x + 1) (x - 1) (2 u^{2} + (- 1 - 6) u + 3) = 0$

Apply the distributive property.

$(x + 1) (x - 1) (2 u^{2} - 1 u - 6 u + 3) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 1) (x - 1) ((2 u^{2} - 1 u) - 6 u + 3) = 0$

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

$(x + 1) (x - 1) (u (2 u - 1) - 3 (2 u - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$(x + 1) (x - 1) ((2 u - 1) (u - 3)) = 0$

Factor.

Replace all occurrences of $u$ with $x$ .

$(x + 1) (x - 1) ((2 x - 1) (x - 3)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x - 1) (2 x - 1) (x - 3) = 0$

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

$x + 1 = 0$

$x - 1 = 0$

$2 x - 1 = 0$

$x - 3 = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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 $1$ .

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

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $(x + 1) (x - 1) (2 x - 1) (x - 3) = 0$ true.

$x = - 1, 1, \frac{1}{2}, 3$

Step 3

Find the Roots (Zeros) V(x)=2x^4-7x^3+x^2+7x-3

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