Find the Roots (Zeros) 28x^3+88x^2-37x-7=0

$28 x^{3} + 88 x^{2} - 37 x - 7 = 0$

Step 1

Factor the left side of the equation.

Factor $28 x^{3} + 88 x^{2} - 37 x - 7$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 7 q = \pm 1, \pm 28, \pm 2, \pm 14, \pm 4, \pm 7$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.03571428, \pm 0.5, \pm 0.0\overline{714285}, \pm 0.25, \pm 0.\overline{142857}, \pm 7, \pm 3.5, \pm 1.75$

Substitute $- 0.\overline{142857}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{142857}$ is a root of the polynomial.

Substitute $- 0.\overline{142857}$ into the polynomial.

$28 {(- 0.\overline{142857})}^{3} + 88 {(- 0.\overline{142857})}^{2} - 37 \cdot - 0.\overline{142857} - 7$

Raise $- 0.\overline{142857}$ to the power of $3$ .

$28 \cdot - 0.00291545 + 88 {(- 0.\overline{142857})}^{2} - 37 \cdot - 0.\overline{142857} - 7$

Multiply $28$ by $- 0.00291545$ .

$- 0.08163265 + 88 {(- 0.\overline{142857})}^{2} - 37 \cdot - 0.\overline{142857} - 7$

Raise $- 0.\overline{142857}$ to the power of $2$ .

$- 0.08163265 + 88 \cdot 0.02040816 - 37 \cdot - 0.\overline{142857} - 7$

Multiply $88$ by $0.02040816$ .

$- 0.08163265 + 1.79591836 - 37 \cdot - 0.\overline{142857} - 7$

Add $- 0.08163265$ and $1.79591836$ .

$1.71428571 - 37 \cdot - 0.\overline{142857} - 7$

Multiply $- 37$ by $- 0.\overline{142857}$ .

$1.71428571 + 5.28571428 - 7$

Add $1.71428571$ and $5.28571428$ .

$7 - 7$

Subtract $7$ from $7$ .

$0$

Since $- 0.\overline{142857}$ is a known root, divide the polynomial by $7 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{28 x^{3} + 88 x^{2} - 37 x - 7}{7 x + 1}$

Divide $28 x^{3} + 88 x^{2} - 37 x - 7$ by $7 x + 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$7 x$

$1$

$28 x^{3}$

$88 x^{2}$

$37 x$

$7$

Divide the highest order term in the dividend $28 x^{3}$ by the highest order term in divisor $7 x$ .

					$4 x^{2}$
	$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			+	$28 x^{3}$	+	$4 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $28 x^{3} + 4 x^{2}$

				$4 x^{2}$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 x^{2}$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$4 x^{2}$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$

Divide the highest order term in the dividend $84 x^{2}$ by the highest order term in divisor $7 x$ .

				$4 x^{2}$	+	$12 x$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$12 x$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					+	$84 x^{2}$	+	$12 x$

The expression needs to be subtracted from the dividend, so change all the signs in $84 x^{2} + 12 x$

				$4 x^{2}$	+	$12 x$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					-	$84 x^{2}$	-	$12 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 x^{2}$	+	$12 x$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					-	$84 x^{2}$	-	$12 x$
							-	$49 x$

Pull the next terms from the original dividend down into the current dividend.

				$4 x^{2}$	+	$12 x$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					-	$84 x^{2}$	-	$12 x$
							-	$49 x$	-	$7$

Divide the highest order term in the dividend $- 49 x$ by the highest order term in divisor $7 x$ .

				$4 x^{2}$	+	$12 x$	-	$7$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					-	$84 x^{2}$	-	$12 x$
							-	$49 x$	-	$7$

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$12 x$	-	$7$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					-	$84 x^{2}$	-	$12 x$
							-	$49 x$	-	$7$
							-	$49 x$	-	$7$

The expression needs to be subtracted from the dividend, so change all the signs in $- 49 x - 7$

				$4 x^{2}$	+	$12 x$	-	$7$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					-	$84 x^{2}$	-	$12 x$
							-	$49 x$	-	$7$
							+	$49 x$	+	$7$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 x^{2}$	+	$12 x$	-	$7$
$7 x$	+	$1$		$28 x^{3}$	+	$88 x^{2}$	-	$37 x$	-	$7$
			-	$28 x^{3}$	-	$4 x^{2}$
					+	$84 x^{2}$	-	$37 x$
					-	$84 x^{2}$	-	$12 x$
							-	$49 x$	-	$7$
							+	$49 x$	+	$7$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$4 x^{2} + 12 x - 7$

Write $28 x^{3} + 88 x^{2} - 37 x - 7$ as a set of factors.

$(7 x + 1) (4 x^{2} + 12 x - 7) = 0$

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 $a \cdot c = 4 \cdot - 7 = - 28$ and whose sum is $b = 12$ .

Factor $12$ out of $12 x$ .

$(7 x + 1) (4 x^{2} + 12 (x) - 7) = 0$

Rewrite $12$ as $- 2$ plus $14$

$(7 x + 1) (4 x^{2} + (- 2 + 14) x - 7) = 0$

Apply the distributive property.

$(7 x + 1) (4 x^{2} - 2 x + 14 x - 7) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(7 x + 1) ((4 x^{2} - 2 x) + 14 x - 7) = 0$

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

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

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

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

Remove unnecessary parentheses.

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

Step 2

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

$7 x + 1 = 0$

$2 x - 1 = 0$

$2 x + 7 = 0$

Step 3

Set $7 x + 1$ equal to $0$ and solve for $x$ .

Set $7 x + 1$ equal to $0$ .

$7 x + 1 = 0$

Solve $7 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$7 x = - 1$

Divide each term in $7 x = - 1$ by $7$ and simplify.

Divide each term in $7 x = - 1$ by $7$ .

$\frac{7 x}{7} = \frac{- 1}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 1}{7}$

Divide $x$ by $1$ .

$x = \frac{- 1}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{7}$

Step 4

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 5

Set $2 x + 7$ equal to $0$ and solve for $x$ .

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Solve $2 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Divide each term in $2 x = - 7$ by $2$ and simplify.

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 6

The final solution is all the values that make $(7 x + 1) (2 x - 1) (2 x + 7) = 0$ true.

$x = - \frac{1}{7}, \frac{1}{2}, - \frac{7}{2}$

Step 7

Find the Roots (Zeros) 28x^3+88x^2-37x-7=0

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