Write as a Set of Linear Factors f(x)=x^5-3x^4+5x^3-5x^2-6x+8

$f (x) = x^{5} - 3 x^{4} + 5 x^{3} - 5 x^{2} - 6 x + 8$

Step 1

Regroup terms.

$f (x) = x^{5} - 5 x^{2} - 6 x - 3 x^{4} + 5 x^{3} + 8$

Step 2

Factor $x$ out of $x^{5} - 5 x^{2} - 6 x$ .

Factor $x$ out of $x^{5}$ .

$f (x) = x \cdot x^{4} - 5 x^{2} - 6 x - 3 x^{4} + 5 x^{3} + 8$

Factor $x$ out of $- 5 x^{2}$ .

$f (x) = x \cdot x^{4} + x (- 5 x) - 6 x - 3 x^{4} + 5 x^{3} + 8$

Factor $x$ out of $- 6 x$ .

$f (x) = x \cdot x^{4} + x (- 5 x) + x \cdot - 6 - 3 x^{4} + 5 x^{3} + 8$

Factor $x$ out of $x \cdot x^{4} + x (- 5 x)$ .

$f (x) = x (x^{4} - 5 x) + x \cdot - 6 - 3 x^{4} + 5 x^{3} + 8$

Factor $x$ out of $x (x^{4} - 5 x) + x \cdot - 6$ .

$f (x) = x (x^{4} - 5 x - 6) - 3 x^{4} + 5 x^{3} + 8$

Step 3

Factor $x^{4} - 5 x - 6$ 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.

$f (x) = p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 6, \pm 2, \pm 3$

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.

$f (x) = {(- 1)}^{4} - 5 \cdot - 1 - 6$

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

$f (x) = 1 - 5 \cdot - 1 - 6$

Multiply $- 5$ by $- 1$ .

$f (x) = 1 + 5 - 6$

Add $1$ and $5$ .

$f (x) = 6 - 6$

Subtract $6$ from $6$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{4} - 5 x - 6}{x + 1}$

Divide $x^{4} - 5 x - 6$ 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 $0$ .

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 6$

$f (x) =$

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

$f (x) =$

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

$f (x) = x^{3} - x^{2} + x - 6$

Write $x^{4} - 5 x - 6$ as a set of factors.

$f (x) = x ((x + 1) (x^{3} - x^{2} + x - 6)) - 3 x^{4} + 5 x^{3} + 8$

Step 4

Factor.

Factor $x^{3} - x^{2} + x - 6$ 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.

$f (x) = p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 6, \pm 2, \pm 3$

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Substitute $2$ into the polynomial.

$f (x) = 2^{3} - 2^{2} + 2 - 6$

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

$f (x) = 8 - 2^{2} + 2 - 6$

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

$f (x) = 8 - 1 \cdot 4 + 2 - 6$

Multiply $- 1$ by $4$ .

$f (x) = 8 - 4 + 2 - 6$

Subtract $4$ from $8$ .

$f (x) = 4 + 2 - 6$

Add $4$ and $2$ .

$f (x) = 6 - 6$

Subtract $6$ from $6$ .

$f (x) = 0$

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

$f (x) = \frac{x^{3} - x^{2} + x - 6}{x - 2}$

Divide $x^{3} - x^{2} + x - 6$ by $x - 2$ .

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x - 6$

$f (x) =$

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

$f (x) =$

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

$f (x) = x^{2} + x + 3$

Write $x^{3} - x^{2} + x - 6$ as a set of factors.

$f (x) = x ((x + 1) ((x - 2) (x^{2} + x + 3))) - 3 x^{4} + 5 x^{3} + 8$

Remove unnecessary parentheses.

$f (x) = x ((x + 1) (x - 2) (x^{2} + x + 3)) - 3 x^{4} + 5 x^{3} + 8$

Remove unnecessary parentheses.

$f (x) = x (x + 1) (x - 2) (x^{2} + x + 3) - 3 x^{4} + 5 x^{3} + 8$

Step 5

Factor $- 3 x^{4} + 5 x^{3} + 8$ 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.

$f (x) = p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1, \pm 3$

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

$f (x) = \pm 1, \pm 0.\overline{3}, \pm 8, \pm 2.\overline{6}, \pm 2, \pm 0.\overline{6}, \pm 4, \pm 1.\overline{3}$

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.

$f (x) = - 3 {(- 1)}^{4} + 5 {(- 1)}^{3} + 8$

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

$f (x) = - 3 \cdot 1 + 5 {(- 1)}^{3} + 8$

Multiply $- 3$ by $1$ .

$f (x) = - 3 + 5 {(- 1)}^{3} + 8$

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

$f (x) = - 3 + 5 \cdot - 1 + 8$

Multiply $5$ by $- 1$ .

$f (x) = - 3 - 5 + 8$

Subtract $5$ from $- 3$ .

$f (x) = - 8 + 8$

Add $- 8$ and $8$ .

$f (x) = 0$

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.

$f (x) = \frac{- 3 x^{4} + 5 x^{3} + 8}{x + 1}$

Divide $- 3 x^{4} + 5 x^{3} + 8$ 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 $0$ .

$f (x) =$

Divide the highest order term in the dividend $- 3 x^{4}$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $8 x^{3}$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} + 8 x^{2}$

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 8 x$

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x + 8$

$f (x) =$

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

$f (x) =$

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

$f (x) = - 3 x^{3} + 8 x^{2} - 8 x + 8$

Write $- 3 x^{4} + 5 x^{3} + 8$ as a set of factors.

$f (x) = x (x + 1) (x - 2) (x^{2} + x + 3) + (x + 1) (- 3 x^{3} + 8 x^{2} - 8 x + 8)$

Step 6

Factor $- 3 x^{3} + 8 x^{2} - 8 x + 8$ 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.

$f (x) = p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1, \pm 3$

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

$f (x) = \pm 1, \pm 0.\overline{3}, \pm 8, \pm 2.\overline{6}, \pm 2, \pm 0.\overline{6}, \pm 4, \pm 1.\overline{3}$

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Substitute $2$ into the polynomial.

$f (x) = - 3 \cdot 2^{3} + 8 \cdot 2^{2} - 8 \cdot 2 + 8$

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

$f (x) = - 3 \cdot 8 + 8 \cdot 2^{2} - 8 \cdot 2 + 8$

Multiply $- 3$ by $8$ .

$f (x) = - 24 + 8 \cdot 2^{2} - 8 \cdot 2 + 8$

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

$f (x) = - 24 + 8 \cdot 4 - 8 \cdot 2 + 8$

Multiply $8$ by $4$ .

$f (x) = - 24 + 32 - 8 \cdot 2 + 8$

Add $- 24$ and $32$ .

$f (x) = 8 - 8 \cdot 2 + 8$

Multiply $- 8$ by $2$ .

$f (x) = 8 - 16 + 8$

Subtract $16$ from $8$ .

$f (x) = - 8 + 8$

Add $- 8$ and $8$ .

$f (x) = 0$

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

$f (x) = \frac{- 3 x^{3} + 8 x^{2} - 8 x + 8}{x - 2}$

Divide $- 3 x^{3} + 8 x^{2} - 8 x + 8$ by $x - 2$ .

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{3} + 6 x^{2}$

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 4 x$

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x + 8$

$f (x) =$

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

$f (x) =$

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

$f (x) = - 3 x^{2} + 2 x - 4$

Write $- 3 x^{3} + 8 x^{2} - 8 x + 8$ as a set of factors.

$f (x) = x (x + 1) (x - 2) (x^{2} + x + 3) + (x + 1) ((x - 2) (- 3 x^{2} + 2 x - 4))$

Remove unnecessary parentheses.

$f (x) = x (x + 1) (x - 2) (x^{2} + x + 3) + (x + 1) (x - 2) (- 3 x^{2} + 2 x - 4)$

Step 7

Factor $(x + 1) (x - 2)$ out of $x (x + 1) (x - 2) (x^{2} + x + 3) + (x + 1) (x - 2) (- 3 x^{2} + 2 x - 4)$ .

Factor $(x + 1) (x - 2)$ out of $x (x + 1) (x - 2) (x^{2} + x + 3)$ .

$f (x) = (x + 1) (x - 2) ((x) (x^{2} + x + 3)) + (x + 1) (x - 2) (- 3 x^{2} + 2 x - 4)$

Factor $(x + 1) (x - 2)$ out of $(x + 1) (x - 2) ((x) (x^{2} + x + 3)) + (x + 1) (x - 2) (- 3 x^{2} + 2 x - 4)$ .

$f (x) = (x + 1) (x - 2) ((x) (x^{2} + x + 3) - 3 x^{2} + 2 x - 4)$

Step 8

Apply the distributive property.

$f (x) = (x + 1) (x - 2) (x \cdot x^{2} + x \cdot x + x \cdot 3 - 3 x^{2} + 2 x - 4)$

Step 9

Simplify.

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

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

$f (x) = (x + 1) (x - 2) (x \cdot x^{2} + x \cdot x + x \cdot 3 - 3 x^{2} + 2 x - 4)$

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

$f (x) = (x + 1) (x - 2) (x^{1 + 2} + x \cdot x + x \cdot 3 - 3 x^{2} + 2 x - 4)$

Add $1$ and $2$ .

$f (x) = (x + 1) (x - 2) (x^{3} + x \cdot x + x \cdot 3 - 3 x^{2} + 2 x - 4)$

Multiply $x$ by $x$ .

$f (x) = (x + 1) (x - 2) (x^{3} + x^{2} + x \cdot 3 - 3 x^{2} + 2 x - 4)$

Move $3$ to the left of $x$ .

$f (x) = (x + 1) (x - 2) (x^{3} + x^{2} + 3 \cdot x - 3 x^{2} + 2 x - 4)$

$f (x) = (x + 1) (x - 2) (x^{3} + x^{2} + 3 x - 3 x^{2} + 2 x - 4)$

Step 10

Subtract $3 x^{2}$ from $x^{2}$ .

$f (x) = (x + 1) (x - 2) (x^{3} - 2 x^{2} + 3 x + 2 x - 4)$

Step 11

Add $3 x$ and $2 x$ .

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

Step 12

Factor.

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

$f (x) = p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 4, \pm 2$

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.

$f (x) = 1^{3} - 2 \cdot 1^{2} + 5 \cdot 1 - 4$

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

$f (x) = 1 - 2 \cdot 1^{2} + 5 \cdot 1 - 4$

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

$f (x) = 1 - 2 \cdot 1 + 5 \cdot 1 - 4$

Multiply $- 2$ by $1$ .

$f (x) = 1 - 2 + 5 \cdot 1 - 4$

Subtract $2$ from $1$ .

$f (x) = - 1 + 5 \cdot 1 - 4$

Multiply $5$ by $1$ .

$f (x) = - 1 + 5 - 4$

Add $- 1$ and $5$ .

$f (x) = 4 - 4$

Subtract $4$ from $4$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} - 2 x^{2} + 5 x - 4}{x - 1}$

Divide $x^{3} - 2 x^{2} + 5 x - 4$ 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 $0$ .

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) = x^{2} - x + 4$

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

$f (x) = (x + 1) (x - 2) ((x - 1) (x^{2} - x + 4))$

Remove unnecessary parentheses.

$f (x) = (x + 1) (x - 2) (x - 1) (x^{2} - x + 4)$

Write as a Set of Linear Factors f(x)=x^5-3x^4+5x^3-5x^2-6x+8

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