Find the Roots/Zeros Using the Rational Roots Test 5p+3(p-8)

$5 p + 3 (p - 8)$

Step 1

Simplify $5 p + 3 (p - 8)$ .

Simplify each term.

Apply the distributive property.

$y = 5 p + 3 p + 3 \cdot - 8$

Multiply $3$ by $- 8$ .

$y = 5 p + 3 p - 24$

Add $5 p$ and $3 p$ .

$y = 8 p - 24$

Step 2

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$

$q = \pm 1, \pm 2, \pm 4, \pm 8$

Step 3

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8}, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$

Step 4

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.

$8 (3) - 24$

Step 5

Simplify the expression. In this case, the expression is equal to $0$ so $p = 3$ is a root of the polynomial.

Multiply $8$ by $3$ .

$24 - 24$

Subtract $24$ from $24$ .

$0$

Step 6

Since $3$ is a known root, divide the polynomial by $p - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{8 p - 24}{p - 3}$

Step 7

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

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

$3$	$8$	$- 24$
	\xa0	\xa0

The first number in the dividend $(8)$ is put into the first position of the result area (below the horizontal line).

$3$	$8$	$- 24$
	\xa0	\xa0
	$8$

Multiply the newest entry in the result $(8)$ by the divisor $(3)$ and place the result of $(24)$ under the next term in the dividend $(- 24)$ .

$3$	$8$	$- 24$
	\xa0	$24$
	$8$

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

$3$	$8$	$- 24$
	\xa0	$24$
	$8$	$0$

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

$8$

Step 8

Since $8 \neq 0$ , there are no solutions.

No solution

Step 9

The polynomial can be written as a set of linear factors.

$p - 3$

Step 10

These are the roots (zeros) of the polynomial $8 p - 24$ .

$p = 3$

Step 11

Simplify $5 p + 3 (p - 8)$ .

Simplify each term.

Apply the distributive property.

$5 p + 3 p + 3 \cdot - 8 = 0$

Multiply $3$ by $- 8$ .

$5 p + 3 p - 24 = 0$

Add $5 p$ and $3 p$ .

$8 p - 24 = 0$

Step 12

Add $24$ to both sides of the equation.

$8 p = 24$

Step 13

Divide each term in $8 p = 24$ by $8$ and simplify.

Divide each term in $8 p = 24$ by $8$ .

$\frac{8 p}{8} = \frac{24}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 p}{8} = \frac{24}{8}$

Divide $p$ by $1$ .

$p = \frac{24}{8}$

Simplify the right side.

Divide $24$ by $8$ .

$p = 3$

Step 14

Find the Roots/Zeros Using the Rational Roots Test 5p+3(p-8)

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