Find the Roots/Zeros Using the Rational Roots Test 60x^4+86x^3-46x^2-43x+8

$60 x^{4} + 86 x^{3} - 46 x^{2} - 43 x + 8$

Step 1

Factor the left side of the equation.

Regroup terms.

$86 x^{3} - 43 x + 60 x^{4} - 46 x^{2} + 8 = 0$

Factor $43 x$ out of $86 x^{3} - 43 x$ .

Factor $43 x$ out of $86 x^{3}$ .

$43 x (2 x^{2}) - 43 x + 60 x^{4} - 46 x^{2} + 8 = 0$

Factor $43 x$ out of $- 43 x$ .

$43 x (2 x^{2}) + 43 x (- 1) + 60 x^{4} - 46 x^{2} + 8 = 0$

Factor $43 x$ out of $43 x (2 x^{2}) + 43 x (- 1)$ .

$43 x (2 x^{2} - 1) + 60 x^{4} - 46 x^{2} + 8 = 0$

Factor $2$ out of $60 x^{4} - 46 x^{2} + 8$ .

Factor $2$ out of $60 x^{4}$ .

$43 x (2 x^{2} - 1) + 2 (30 x^{4}) - 46 x^{2} + 8 = 0$

Factor $2$ out of $- 46 x^{2}$ .

$43 x (2 x^{2} - 1) + 2 (30 x^{4}) + 2 (- 23 x^{2}) + 8 = 0$

Factor $2$ out of $8$ .

$43 x (2 x^{2} - 1) + 2 (30 x^{4}) + 2 (- 23 x^{2}) + 2 (4) = 0$

Factor $2$ out of $2 (30 x^{4}) + 2 (- 23 x^{2})$ .

$43 x (2 x^{2} - 1) + 2 (30 x^{4} - 23 x^{2}) + 2 (4) = 0$

Factor $2$ out of $2 (30 x^{4} - 23 x^{2}) + 2 (4)$ .

$43 x (2 x^{2} - 1) + 2 (30 x^{4} - 23 x^{2} + 4) = 0$

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

$43 x (2 x^{2} - 1) + 2 (30 {(x^{2})}^{2} - 23 x^{2} + 4) = 0$

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

$43 x (2 x^{2} - 1) + 2 (30 u^{2} - 23 u + 4) = 0$

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 $a \cdot c = 30 \cdot 4 = 120$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 u$ .

$43 x (2 x^{2} - 1) + 2 (30 u^{2} - 23 u + 4) = 0$

Rewrite $- 23$ as $- 8$ plus $- 15$

$43 x (2 x^{2} - 1) + 2 (30 u^{2} + (- 8 - 15) u + 4) = 0$

Apply the distributive property.

$43 x (2 x^{2} - 1) + 2 (30 u^{2} - 8 u - 15 u + 4) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$43 x (2 x^{2} - 1) + 2 ((30 u^{2} - 8 u) - 15 u + 4) = 0$

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

$43 x (2 x^{2} - 1) + 2 (2 u (15 u - 4) - (15 u - 4)) = 0$

Factor the polynomial by factoring out the greatest common factor, $15 u - 4$ .

$43 x (2 x^{2} - 1) + 2 ((15 u - 4) (2 u - 1)) = 0$

Factor.

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

$43 x (2 x^{2} - 1) + 2 ((15 x^{2} - 4) (2 x^{2} - 1)) = 0$

Remove unnecessary parentheses.

$43 x (2 x^{2} - 1) + 2 (15 x^{2} - 4) (2 x^{2} - 1) = 0$

Factor $2 x^{2} - 1$ out of $43 x (2 x^{2} - 1) + 2 (15 x^{2} - 4) (2 x^{2} - 1)$ .

Factor $2 x^{2} - 1$ out of $43 x (2 x^{2} - 1)$ .

$(2 x^{2} - 1) (43 x) + 2 (15 x^{2} - 4) (2 x^{2} - 1) = 0$

Factor $2 x^{2} - 1$ out of $2 (15 x^{2} - 4) (2 x^{2} - 1)$ .

$(2 x^{2} - 1) (43 x) + (2 x^{2} - 1) (2 (15 x^{2} - 4)) = 0$

Factor $2 x^{2} - 1$ out of $(2 x^{2} - 1) (43 x) + (2 x^{2} - 1) (2 (15 x^{2} - 4))$ .

$(2 x^{2} - 1) (43 x + 2 (15 x^{2} - 4)) = 0$

Apply the distributive property.

$(2 x^{2} - 1) (43 x + 2 (15 x^{2}) + 2 \cdot - 4) = 0$

Multiply $15$ by $2$ .

$(2 x^{2} - 1) (43 x + 30 x^{2} + 2 \cdot - 4) = 0$

Multiply $2$ by $- 4$ .

$(2 x^{2} - 1) (43 x + 30 x^{2} - 8) = 0$

Reorder terms.

$(2 x^{2} - 1) (30 x^{2} + 43 x - 8) = 0$

Factor.

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 $a \cdot c = 30 \cdot - 8 = - 240$ and whose sum is $b = 43$ .

Factor $43$ out of $43 x$ .

$(2 x^{2} - 1) (30 x^{2} + 43 (x) - 8) = 0$

Rewrite $43$ as $- 5$ plus $48$

$(2 x^{2} - 1) (30 x^{2} + (- 5 + 48) x - 8) = 0$

Apply the distributive property.

$(2 x^{2} - 1) (30 x^{2} - 5 x + 48 x - 8) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} - 1) ((30 x^{2} - 5 x) + 48 x - 8) = 0$

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

$(2 x^{2} - 1) (5 x (6 x - 1) + 8 (6 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $6 x - 1$ .

$(2 x^{2} - 1) ((6 x - 1) (5 x + 8)) = 0$

Remove unnecessary parentheses.

$(2 x^{2} - 1) (6 x - 1) (5 x + 8) = 0$

Step 2

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

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

$6 x - 1 = 0$

$5 x + 8 = 0$

Step 3

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

Set $2 x^{2} - 1$ equal to $0$ .

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

Solve $2 x^{2} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x^{2} = 1$

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

Divide each term in $2 x^{2} = 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x^{2}$ by $1$ .

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

Take the square root of both sides of the equation to eliminate the exponent on the left side.

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

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

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

Next, use the negative value of the $\pm$ to find the second solution.

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

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 4

Set $6 x - 1$ equal to $0$ and solve for $x$ .

Set $6 x - 1$ equal to $0$ .

$6 x - 1 = 0$

Solve $6 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$6 x = 1$

Divide each term in $6 x = 1$ by $6$ and simplify.

Divide each term in $6 x = 1$ by $6$ .

$\frac{6 x}{6} = \frac{1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{1}{6}$

Divide $x$ by $1$ .

$x = \frac{1}{6}$

Step 5

Set $5 x + 8$ equal to $0$ and solve for $x$ .

Set $5 x + 8$ equal to $0$ .

$5 x + 8 = 0$

Solve $5 x + 8 = 0$ for $x$ .

Subtract $8$ from both sides of the equation.

$5 x = - 8$

Divide each term in $5 x = - 8$ by $5$ and simplify.

Divide each term in $5 x = - 8$ by $5$ .

$\frac{5 x}{5} = \frac{- 8}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 8}{5}$

Divide $x$ by $1$ .

$x = \frac{- 8}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{8}{5}$

Step 6

The final solution is all the values that make $(2 x^{2} - 1) (6 x - 1) (5 x + 8) = 0$ true.

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}, \frac{1}{6}, - \frac{8}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}, \frac{1}{6}, - \frac{8}{5}$

Decimal Form:

$x = 0.70710678 \dots, - 0.70710678 \dots, 0.1\overline{6}, - 1.6$

Mixed Number Form:

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}, \frac{1}{6}, - 1 \frac{3}{5}$

Step 8

Find the Roots/Zeros Using the Rational Roots Test 60x^4+86x^3-46x^2-43x+8

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