Algebra Examples

$2 x^{3} - 23 x^{2} + 58 x + 35$

Step 1

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.

$p = \pm 1, \pm 5, \pm 7, \pm 35$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2}, \pm 7, \pm \frac{7}{2}, \pm 35, \pm \frac{35}{2}$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$2 {(- \frac{1}{2})}^{3} - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - \frac{1}{2}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

$2 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.1.2

Apply the product rule to $\frac{1}{2}$ .

$2 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.2

Raise $- 1$ to the power of $3$ .

$2 (- \frac{1^{3}}{2^{3}}) - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.3

One to any power is one.

$2 (- \frac{1}{2^{3}}) - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.4

Raise $2$ to the power of $3$ .

$2 (- \frac{1}{8}) - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.5

Cancel the common factor of $2$ .

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Step 4.1.5.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$2 (\frac{- 1}{8}) - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.5.2

Factor $2$ out of $8$ .

$2 \frac{- 1}{2 (4)} - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.5.3

Cancel the common factor.

$2 \frac{- 1}{2 \cdot 4} - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.5.4

Rewrite the expression.

$\frac{- 1}{4} - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.6

Move the negative in front of the fraction.

$- \frac{1}{4} - 23 {(- \frac{1}{2})}^{2} + 58 (- \frac{1}{2}) + 35$

Step 4.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.7.1

Apply the product rule to $- \frac{1}{2}$ .

$- \frac{1}{4} - 23 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) + 58 (- \frac{1}{2}) + 35$

Step 4.1.7.2

Apply the product rule to $\frac{1}{2}$ .

$- \frac{1}{4} - 23 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) + 58 (- \frac{1}{2}) + 35$

Step 4.1.8

Raise $- 1$ to the power of $2$ .

$- \frac{1}{4} - 23 (1 \frac{1^{2}}{2^{2}}) + 58 (- \frac{1}{2}) + 35$

Step 4.1.9

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

$- \frac{1}{4} - 23 \frac{1^{2}}{2^{2}} + 58 (- \frac{1}{2}) + 35$

Step 4.1.10

One to any power is one.

$- \frac{1}{4} - 23 \frac{1}{2^{2}} + 58 (- \frac{1}{2}) + 35$

Step 4.1.11

Raise $2$ to the power of $2$ .

$- \frac{1}{4} - 23 (\frac{1}{4}) + 58 (- \frac{1}{2}) + 35$

Step 4.1.12

Combine $- 23$ and $\frac{1}{4}$ .

$- \frac{1}{4} + \frac{- 23}{4} + 58 (- \frac{1}{2}) + 35$

Step 4.1.13

Move the negative in front of the fraction.

$- \frac{1}{4} - \frac{23}{4} + 58 (- \frac{1}{2}) + 35$

Step 4.1.14

Cancel the common factor of $2$ .

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Step 4.1.14.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{1}{4} - \frac{23}{4} + 58 (\frac{- 1}{2}) + 35$

Step 4.1.14.2

Factor $2$ out of $58$ .

$- \frac{1}{4} - \frac{23}{4} + 2 (29) \frac{- 1}{2} + 35$

Step 4.1.14.3

Cancel the common factor.

$- \frac{1}{4} - \frac{23}{4} + 2 \cdot 29 \frac{- 1}{2} + 35$

Step 4.1.14.4

Rewrite the expression.

$- \frac{1}{4} - \frac{23}{4} + 29 \cdot - 1 + 35$

Step 4.1.15

Multiply $29$ by $- 1$ .

$- \frac{1}{4} - \frac{23}{4} - 29 + 35$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 29 + 35 + \frac{- 1 - 23}{4}$

Step 4.2.2

Subtract $23$ from $- 1$ .

$- 29 + 35 + \frac{- 24}{4}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 29$ as a fraction with denominator $1$ .

$\frac{- 29}{1} + 35 + \frac{- 24}{4}$

Step 4.3.2

Multiply $\frac{- 29}{1}$ by $\frac{4}{4}$ .

$\frac{- 29}{1} \cdot \frac{4}{4} + 35 + \frac{- 24}{4}$

Step 4.3.3

Multiply $\frac{- 29}{1}$ by $\frac{4}{4}$ .

$\frac{- 29 \cdot 4}{4} + 35 + \frac{- 24}{4}$

Step 4.3.4

Write $35$ as a fraction with denominator $1$ .

$\frac{- 29 \cdot 4}{4} + \frac{35}{1} + \frac{- 24}{4}$

Step 4.3.5

Multiply $\frac{35}{1}$ by $\frac{4}{4}$ .

$\frac{- 29 \cdot 4}{4} + \frac{35}{1} \cdot \frac{4}{4} + \frac{- 24}{4}$

Step 4.3.6

Multiply $\frac{35}{1}$ by $\frac{4}{4}$ .

$\frac{- 29 \cdot 4}{4} + \frac{35 \cdot 4}{4} + \frac{- 24}{4}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 29 \cdot 4 + 35 \cdot 4 - 24}{4}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 29$ by $4$ .

$\frac{- 116 + 35 \cdot 4 - 24}{4}$

Step 4.5.2

Multiply $35$ by $4$ .

$\frac{- 116 + 140 - 24}{4}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $- 116$ and $140$ .

$\frac{24 - 24}{4}$

Step 4.6.2

Subtract $24$ from $24$ .

$\frac{0}{4}$

Step 4.6.3

Divide $0$ by $4$ .

$0$

Step 5

Since $- \frac{1}{2}$ is a known root, divide the polynomial by $x + \frac{1}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 23 x^{2} + 58 x + 35}{x + \frac{1}{2}}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 1)$ under the next term in the dividend $(- 23)$ .

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$
		$- 1$
	$2$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$
		$- 1$
	$2$	$- 24$

Step 6.5

Multiply the newest entry in the result $(- 24)$ by the divisor $(- \frac{1}{2})$ and place the result of $(12)$ under the next term in the dividend $(58)$ .

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$
		$- 1$	$12$
	$2$	$- 24$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$
		$- 1$	$12$
	$2$	$- 24$	$70$

Step 6.7

Multiply the newest entry in the result $(70)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 35)$ under the next term in the dividend $(35)$ .

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$
		$- 1$	$12$	$- 35$
	$2$	$- 24$	$70$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{1}{2}$	$2$	$- 23$	$58$	$35$
		$- 1$	$12$	$- 35$
	$2$	$- 24$	$70$	$0$

Step 6.9

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$2 x^{2} + - 24 x + 70$

Step 6.10

Simplify the quotient polynomial.

$2 x^{2} - 24 x + 70$

Step 7

Factor $2$ out of $2 x^{2} - 24 x + 70$ .

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Step 7.1

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

$(x + \frac{1}{2}) (2 (x^{2}) - 24 x + 70)$

Step 7.2

Factor $2$ out of $- 24 x$ .

$(x + \frac{1}{2}) (2 (x^{2}) + 2 (- 12 x) + 70)$

Step 7.3

Factor $2$ out of $70$ .

$(x + \frac{1}{2}) (2 x^{2} + 2 (- 12 x) + 2 \cdot 35)$

Step 7.4

Factor $2$ out of $2 x^{2} + 2 (- 12 x)$ .

$(x + \frac{1}{2}) (2 (x^{2} - 12 x) + 2 \cdot 35)$

Step 7.5

Factor $2$ out of $2 (x^{2} - 12 x) + 2 \cdot 35$ .

$(x + \frac{1}{2}) (2 (x^{2} - 12 x + 35))$

Step 8

Factor.

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Step 8.1

Factor $x^{2} - 12 x + 35$ using the AC method.

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Step 8.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $35$ and whose sum is $- 12$ .

$- 7, - 5$

Step 8.1.2

Write the factored form using these integers.

$(x + \frac{1}{2}) (2 ((x - 7) (x - 5)))$

Step 8.2

Remove unnecessary parentheses.

$(x + \frac{1}{2}) (2 (x - 7) (x - 5))$

Step 9

Factor the left side of the equation.

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Step 9.1

Factor $2 x^{3} - 23 x^{2} + 58 x + 35$ using the rational roots test.

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Step 9.1.1

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.

$p = \pm 1, \pm 35, \pm 5, \pm 7$

$q = \pm 1, \pm 2$

Step 9.1.2

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

$\pm 1, \pm 0.5, \pm 35, \pm 17.5, \pm 5, \pm 2.5, \pm 7, \pm 3.5$

Step 9.1.3

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

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Step 9.1.3.1

Substitute $- 0.5$ into the polynomial.

$2 {(- 0.5)}^{3} - 23 {(- 0.5)}^{2} + 58 \cdot - 0.5 + 35$

Step 9.1.3.2

Raise $- 0.5$ to the power of $3$ .

$2 \cdot - 0.125 - 23 {(- 0.5)}^{2} + 58 \cdot - 0.5 + 35$

Step 9.1.3.3

Multiply $2$ by $- 0.125$ .

$- 0.25 - 23 {(- 0.5)}^{2} + 58 \cdot - 0.5 + 35$

Step 9.1.3.4

Raise $- 0.5$ to the power of $2$ .

$- 0.25 - 23 \cdot 0.25 + 58 \cdot - 0.5 + 35$

Step 9.1.3.5

Multiply $- 23$ by $0.25$ .

$- 0.25 - 5.75 + 58 \cdot - 0.5 + 35$

Step 9.1.3.6

Subtract $5.75$ from $- 0.25$ .

$- 6 + 58 \cdot - 0.5 + 35$

Step 9.1.3.7

Multiply $58$ by $- 0.5$ .

$- 6 - 29 + 35$

Step 9.1.3.8

Subtract $29$ from $- 6$ .

$- 35 + 35$

Step 9.1.3.9

Add $- 35$ and $35$ .

$0$

Step 9.1.4

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 23 x^{2} + 58 x + 35}{2 x + 1}$

Step 9.1.5

Divide $2 x^{3} - 23 x^{2} + 58 x + 35$ by $2 x + 1$ .

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Step 9.1.5.1

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

$2 x$

$1$

$2 x^{3}$

$23 x^{2}$

$58 x$

$35$

Step 9.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $2 x$ .

					$x^{2}$
	$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$

Step 9.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			+	$2 x^{3}$	+	$x^{2}$

Step 9.1.5.4

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$

Step 9.1.5.5

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$

Step 9.1.5.6

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$

Step 9.1.5.7

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$12 x$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$

Step 9.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$12 x$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					-	$24 x^{2}$	-	$12 x$

Step 9.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} - 12 x$

				$x^{2}$	-	$12 x$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					+	$24 x^{2}$	+	$12 x$

Step 9.1.5.10

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

				$x^{2}$	-	$12 x$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					+	$24 x^{2}$	+	$12 x$
							+	$70 x$

Step 9.1.5.11

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

				$x^{2}$	-	$12 x$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					+	$24 x^{2}$	+	$12 x$
							+	$70 x$	+	$35$

Step 9.1.5.12

Divide the highest order term in the dividend $70 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$12 x$	+	$35$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					+	$24 x^{2}$	+	$12 x$
							+	$70 x$	+	$35$

Step 9.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$12 x$	+	$35$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					+	$24 x^{2}$	+	$12 x$
							+	$70 x$	+	$35$
							+	$70 x$	+	$35$

Step 9.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $70 x + 35$

				$x^{2}$	-	$12 x$	+	$35$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					+	$24 x^{2}$	+	$12 x$
							+	$70 x$	+	$35$
							-	$70 x$	-	$35$

Step 9.1.5.15

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

				$x^{2}$	-	$12 x$	+	$35$
$2 x$	+	$1$		$2 x^{3}$	-	$23 x^{2}$	+	$58 x$	+	$35$
			-	$2 x^{3}$	-	$x^{2}$
					-	$24 x^{2}$	+	$58 x$
					+	$24 x^{2}$	+	$12 x$
							+	$70 x$	+	$35$
							-	$70 x$	-	$35$
										$0$

Step 9.1.5.16

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

$x^{2} - 12 x + 35$

Step 9.1.6

Write $2 x^{3} - 23 x^{2} + 58 x + 35$ as a set of factors.

$(2 x + 1) (x^{2} - 12 x + 35) = 0$

Step 9.2

Factor $x^{2} - 12 x + 35$ using the AC method.

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Step 9.2.1

Factor $x^{2} - 12 x + 35$ using the AC method.

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Step 9.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $35$ and whose sum is $- 12$ .

$- 7, - 5$

Step 9.2.1.2

Write the factored form using these integers.

$(2 x + 1) ((x - 7) (x - 5)) = 0$

Step 9.2.2

Remove unnecessary parentheses.

$(2 x + 1) (x - 7) (x - 5) = 0$

Step 10

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

$2 x + 1 = 0$

$x - 7 = 0$

$x - 5 = 0$

Step 11

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

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Step 11.1

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

$2 x + 1 = 0$

Step 11.2

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

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Step 11.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 11.2.2

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

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Step 11.2.2.1

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

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

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

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

Step 11.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 11.2.2.3

Simplify the right side.

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Step 11.2.2.3.1

Move the negative in front of the fraction.

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

Step 12

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 12.2

Add $7$ to both sides of the equation.

$x = 7$

Step 13

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 13.2

Add $5$ to both sides of the equation.

$x = 5$

Step 14

The final solution is all the values that make $(2 x + 1) (x - 7) (x - 5) = 0$ true.

$x = - \frac{1}{2}, 7, 5$

Step 15