Write as a Set of Linear Factors x^5+4x^3-17x+12

$x^{5} + 4 x^{3} - 17 x + 12$

Step 1

Factor $x^{5} + 4 x^{3} - 17 x + 12$ 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.

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

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

$1^{5} + 4 \cdot 1^{3} - 17 \cdot 1 + 12$

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

$1 + 4 \cdot 1^{3} - 17 \cdot 1 + 12$

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

$1 + 4 \cdot 1 - 17 \cdot 1 + 12$

Multiply $4$ by $1$ .

$1 + 4 - 17 \cdot 1 + 12$

Add $1$ and $4$ .

$5 - 17 \cdot 1 + 12$

Multiply $- 17$ by $1$ .

$5 - 17 + 12$

Subtract $17$ from $5$ .

$- 12 + 12$

Add $- 12$ and $12$ .

$0$

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

$\frac{x^{5} + 4 x^{3} - 17 x + 12}{x - 1}$

Divide $x^{5} + 4 x^{3} - 17 x + 12$ by $x - 1$ .

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

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

Divide the highest order term in the dividend $x^{5}$ by the highest order term in divisor $x$ .

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} - x^{4}$

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

$x^{4}$

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 5 x^{2}$

$x^{4}$

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

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

$x^{4}$

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

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

$x^{4}$

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x$ .

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} - 5 x$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

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

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

$12 x$

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

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

$12 x$

$12$

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

$12 x$

$12$

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

$12 x$

$12$

$12 x$

$12$

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x + 12$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

$12 x$

$12$

$12 x$

$12$

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

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x$

$1$

$x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$17 x$

$12$

$x^{5}$

$x^{4}$

$4 x^{3}$

$x^{4}$

$x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$17 x$

$5 x^{2}$

$5 x$

$12 x$

$12$

$12 x$

$12$

$0$

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

$x^{4} + x^{3} + 5 x^{2} + 5 x - 12$

Write $x^{5} + 4 x^{3} - 17 x + 12$ as a set of factors.

$(x - 1) (x^{4} + x^{3} + 5 x^{2} + 5 x - 12)$

Step 2

Factor $x^{4} + x^{3} + 5 x^{2} + 5 x - 12$ 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.

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

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

$1^{4} + 1^{3} + 5 \cdot 1^{2} + 5 \cdot 1 - 12$

Raise $1$ to the power of $4$ .

$1 + 1^{3} + 5 \cdot 1^{2} + 5 \cdot 1 - 12$

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

$1 + 1 + 5 \cdot 1^{2} + 5 \cdot 1 - 12$

Add $1$ and $1$ .

$2 + 5 \cdot 1^{2} + 5 \cdot 1 - 12$

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

$2 + 5 \cdot 1 + 5 \cdot 1 - 12$

Multiply $5$ by $1$ .

$2 + 5 + 5 \cdot 1 - 12$

Add $2$ and $5$ .

$7 + 5 \cdot 1 - 12$

Multiply $5$ by $1$ .

$7 + 5 - 12$

Add $7$ and $5$ .

$12 - 12$

Subtract $12$ from $12$ .

$0$

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

$\frac{x^{4} + x^{3} + 5 x^{2} + 5 x - 12}{x - 1}$

Divide $x^{4} + x^{3} + 5 x^{2} + 5 x - 12$ by $x - 1$ .

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

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

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

$x^{3}$

$2 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} - 7 x$

$x^{3}$

$2 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

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

$x^{3}$

$2 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$12 x$

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

$x^{3}$

$2 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$12 x$

$12$

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$7 x$

$12$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$12 x$

$12$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$7 x$

$12$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$12 x$

$12$

$12 x$

$12$

The expression needs to be subtracted from the dividend, so change all the signs in $12 x - 12$

$x^{3}$

$2 x^{2}$

$7 x$

$12$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$12 x$

$12$

$12 x$

$12$

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

$x^{3}$

$2 x^{2}$

$7 x$

$12$

$x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$5 x$

$12$

$x^{4}$

$x^{3}$

$2 x^{3}$

$5 x^{2}$

$2 x^{3}$

$2 x^{2}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$12 x$

$12$

$12 x$

$12$

$0$

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

$x^{3} + 2 x^{2} + 7 x + 12$

Write $x^{4} + x^{3} + 5 x^{2} + 5 x - 12$ as a set of factors.

$(x - 1) ((x - 1) (x^{3} + 2 x^{2} + 7 x + 12))$

Remove unnecessary parentheses.

$(x - 1) (x - 1) (x^{3} + 2 x^{2} + 7 x + 12)$

Step 3

Combine like factors.

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

${(x - 1)}^{1} (x - 1) (x^{3} + 2 x^{2} + 7 x + 12)$

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

${(x - 1)}^{1} {(x - 1)}^{1} (x^{3} + 2 x^{2} + 7 x + 12)$

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

${(x - 1)}^{1 + 1} (x^{3} + 2 x^{2} + 7 x + 12)$

Add $1$ and $1$ .

${(x - 1)}^{2} (x^{3} + 2 x^{2} + 7 x + 12)$

Write as a Set of Linear Factors x^5+4x^3-17x+12

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