Find the Roots/Zeros Using the Rational Roots Test 5x^5-51x^4+140x^3-76x^2+135x-25

$5 x^{5} - 51 x^{4} + 140 x^{3} - 76 x^{2} + 135 x - 25$

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.

$p = \pm 1, \pm 5, \pm 25$

$q = \pm 1, \pm 5$

Step 2

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

$\pm 1, \pm \frac{1}{5}, \pm 5, \pm 25$

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.

$5 {(5)}^{5} - 51 {(5)}^{4} + 140 {(5)}^{3} - 76 {(5)}^{2} + 135 (5) - 25$

Step 4

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

Simplify each term.

Multiply $5$ by ${(5)}^{5}$ by adding the exponents.

Multiply $5$ by ${(5)}^{5}$ .

Raise $5$ to the power of $1$ .

$5^{1} {(5)}^{5} - 51 {(5)}^{4} + 140 {(5)}^{3} - 76 {(5)}^{2} + 135 (5) - 25$

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

$5^{1 + 5} - 51 {(5)}^{4} + 140 {(5)}^{3} - 76 {(5)}^{2} + 135 (5) - 25$

Add $1$ and $5$ .

$5^{6} - 51 {(5)}^{4} + 140 {(5)}^{3} - 76 {(5)}^{2} + 135 (5) - 25$

Raise $5$ to the power of $6$ .

$15625 - 51 {(5)}^{4} + 140 {(5)}^{3} - 76 {(5)}^{2} + 135 (5) - 25$

Raise $5$ to the power of $4$ .

$15625 - 51 \cdot 625 + 140 {(5)}^{3} - 76 {(5)}^{2} + 135 (5) - 25$

Multiply $- 51$ by $625$ .

$15625 - 31875 + 140 {(5)}^{3} - 76 {(5)}^{2} + 135 (5) - 25$

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

$15625 - 31875 + 140 \cdot 125 - 76 {(5)}^{2} + 135 (5) - 25$

Multiply $140$ by $125$ .

$15625 - 31875 + 17500 - 76 {(5)}^{2} + 135 (5) - 25$

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

$15625 - 31875 + 17500 - 76 \cdot 25 + 135 (5) - 25$

Multiply $- 76$ by $25$ .

$15625 - 31875 + 17500 - 1900 + 135 (5) - 25$

Multiply $135$ by $5$ .

$15625 - 31875 + 17500 - 1900 + 675 - 25$

Simplify by adding and subtracting.

Subtract $31875$ from $15625$ .

$- 16250 + 17500 - 1900 + 675 - 25$

Add $- 16250$ and $17500$ .

$1250 - 1900 + 675 - 25$

Subtract $1900$ from $1250$ .

$- 650 + 675 - 25$

Add $- 650$ and $675$ .

$25 - 25$

Subtract $25$ from $25$ .

$0$

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.

$\frac{5 x^{5} - 51 x^{4} + 140 x^{3} - 76 x^{2} + 135 x - 25}{x - 5}$

Step 6

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.

$5$	$5$	$- 51$	$140$	$- 76$	$135$	$- 25$
	\xa0					\xa0

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$5$	$5$	$- 51$	$140$	$- 76$	$135$	$- 25$
	\xa0					\xa0
	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(5)$ and place the result of $(25)$ under the next term in the dividend $(- 51)$ .

$5$	$5$	$- 51$	$140$	$- 76$	$135$	$- 25$
	\xa0	$25$				\xa0
	$5$