Precalculus Examples

$3 x^{3} - 7 x^{2} - 75 x + 175$

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.

$p = \pm 1, \pm 5, \pm 7, \pm 25, \pm 35, \pm 175$

$q = \pm 1, \pm 3$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm 5, \pm \frac{5}{3}, \pm 7, \pm \frac{7}{3}, \pm 25, \pm \frac{25}{3}, \pm 35, \pm \frac{35}{3}, \pm 175, \pm \frac{175}{3}$

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.

$3 {(5)}^{3} - 7 {(5)}^{2} - 75 \cdot 5 + 175$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 5$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $5$ to the power of $3$ .

$3 \cdot 125 - 7 {(5)}^{2} - 75 \cdot 5 + 175$

Step 4.1.2

Multiply $3$ by $125$ .

$375 - 7 {(5)}^{2} - 75 \cdot 5 + 175$

Step 4.1.3

Raise $5$ to the power of $2$ .

$375 - 7 \cdot 25 - 75 \cdot 5 + 175$

Step 4.1.4

Multiply $- 7$ by $25$ .

$375 - 175 - 75 \cdot 5 + 175$

Step 4.1.5

Multiply $- 75$ by $5$ .

$375 - 175 - 375 + 175$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $175$ from $375$ .

$200 - 375 + 175$

Step 4.2.2

Subtract $375$ from $200$ .

$- 175 + 175$

Step 4.2.3

Add $- 175$ and $175$ .

$0$

Step 5

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} - 7 x^{2} - 75 x + 175}{x - 5}$

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.

$5$	$3$	$- 7$	$- 75$	$175$

Step 6.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$5$	$3$	$- 7$	$- 75$	$175$

	$3$

Step 6.3

Multiply the newest entry in the result $(3)$ by the divisor $(5)$ and place the result of $(15)$ under the next term in the dividend $(- 7)$ .

$5$	$3$	$- 7$	$- 75$	$175$
		$15$
	$3$

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.

$5$	$3$	$- 7$	$- 75$	$175$
		$15$
	$3$	$8$

Step 6.5

Multiply the newest entry in the result $(8)$ by the divisor $(5)$ and place the result of $(40)$ under the next term in the dividend $(- 75)$ .

$5$	$3$	$- 7$	$- 75$	$175$
		$15$	$40$
	$3$	$8$

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.

$5$	$3$	$- 7$	$- 75$	$175$
		$15$	$40$
	$3$	$8$	$- 35$

Step 6.7

Multiply the newest entry in the result $(- 35)$ by the divisor $(5)$ and place the result of $(- 175)$ under the next term in the dividend $(175)$ .

$5$	$3$	$- 7$	$- 75$	$175$
		$15$	$40$	$- 175$
	$3$	$8$	$- 35$

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.

$5$	$3$	$- 7$	$- 75$	$175$
		$15$	$40$	$- 175$
	$3$	$8$	$- 35$	$0$

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.

$3 x^{2} + 8 x - 35$

Step 7

Factor by grouping.

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Step 7.1

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 $a \cdot c = 3 \cdot - 35 = - 105$ and whose sum is $b = 8$ .

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Step 7.1.1

Factor $8$ out of $8 x$ .

$(x - 5) + 3 x^{2} + 8 (x) - 35$

Step 7.1.2

Rewrite $8$ as $- 7$ plus $15$

$(x - 5) + 3 x^{2} + (- 7 + 15) x - 35$

Step 7.1.3

Apply the distributive property.

$(x - 5) + 3 x^{2} - 7 x + 15 x - 35$

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

$(x - 5) (3 x^{2} - 7 x) + 15 x - 35$

Step 7.2.2

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

$(x - 5) + x (3 x - 7) + 5 (3 x - 7)$

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 7$ .

$(x - 5) + (3 x - 7) (x + 5)$

$(x - 5) (3 x - 7) (x + 5)$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor out the greatest common factor from each group.

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Step 8.1.1

Group the first two terms and the last two terms.

$(3 x^{3} - 7 x^{2}) - 75 x + 175 = 0$

Step 8.1.2

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

$x^{2} (3 x - 7) - 25 (3 x - 7) = 0$

Step 8.2

Factor the polynomial by factoring out the greatest common factor, $3 x - 7$ .

$(3 x - 7) (x^{2} - 25) = 0$

Step 8.3

Rewrite $25$ as $5^{2}$ .

$(3 x - 7) (x^{2} - 5^{2}) = 0$

Step 8.4

Factor.

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Step 8.4.1

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 $b = 5$ .

$(3 x - 7) ((x + 5) (x - 5)) = 0$

Step 8.4.2

Remove unnecessary parentheses.

$(3 x - 7) (x + 5) (x - 5) = 0$

Step 9

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

$3 x - 7 = 0$

$x + 5 = 0$

$x - 5 = 0$

Step 10

Set $3 x - 7$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $3 x - 7$ equal to $0$ .

$3 x - 7 = 0$

Step 10.2

Solve $3 x - 7 = 0$ for $x$ .

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Step 10.2.1

Add $7$ to both sides of the equation.

$3 x = 7$

Step 10.2.2

Divide each term in $3 x = 7$ by $3$ and simplify.

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Step 10.2.2.1

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $3$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{3}$

Step 11

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

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Step 11.1

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

$x + 5 = 0$

Step 11.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 12

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

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Step 12.1

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

$x - 5 = 0$

Step 12.2

Add $5$ to both sides of the equation.

$x = 5$

Step 13

The final solution is all the values that make $(3 x - 7) (x + 5) (x - 5) = 0$ true.

$x = \frac{7}{3}, - 5, 5$

Step 14