Write as a Set of Linear Factors f(x)=x^4-10x^3-64x^2-86x-33

$f (x) = x^{4} - 10 x^{3} - 64 x^{2} - 86 x - 33$

Step 1

Factor $x^{4} - 10 x^{3} - 64 x^{2} - 86 x - 33$ 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.

$f (x) = p = \pm 1, \pm 33, \pm 3, \pm 11$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 33, \pm 3, \pm 11$

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.

$f (x) = {(- 1)}^{4} - 10 {(- 1)}^{3} - 64 {(- 1)}^{2} - 86 \cdot - 1 - 33$

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

$f (x) = 1 - 10 {(- 1)}^{3} - 64 {(- 1)}^{2} - 86 \cdot - 1 - 33$

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

$f (x) = 1 - 10 \cdot - 1 - 64 {(- 1)}^{2} - 86 \cdot - 1 - 33$

Multiply $- 10$ by $- 1$ .

$f (x) = 1 + 10 - 64 {(- 1)}^{2} - 86 \cdot - 1 - 33$

Add $1$ and $10$ .

$f (x) = 11 - 64 {(- 1)}^{2} - 86 \cdot - 1 - 33$

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

$f (x) = 11 - 64 \cdot 1 - 86 \cdot - 1 - 33$

Multiply $- 64$ by $1$ .

$f (x) = 11 - 64 - 86 \cdot - 1 - 33$

Subtract $64$ from $11$ .

$f (x) = - 53 - 86 \cdot - 1 - 33$

Multiply $- 86$ by $- 1$ .

$f (x) = - 53 + 86 - 33$

Add $- 53$ and $86$ .

$f (x) = 33 - 33$

Subtract $33$ from $33$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{4} - 10 x^{3} - 64 x^{2} - 86 x - 33}{x + 1}$

Divide $x^{4} - 10 x^{3} - 64 x^{2} - 86 x - 33$ 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 $- 11 x^{3}$ 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 $- 11 x^{3} - 11 x^{2}$

$f (x) =$

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 $- 53 x^{2}$ 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 $- 53 x^{2} - 53 x$

$f (x) =$

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 $- 33 x$ 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 $- 33 x - 33$

$f (x) =$

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.

$f (x) = x^{3} - 11 x^{2} - 53 x - 33$

Write $x^{4} - 10 x^{3} - 64 x^{2} - 86 x - 33$ as a set of factors.

$f (x) = (x + 1) (x^{3} - 11 x^{2} - 53 x - 33)$

Step 2

Factor $x^{3} - 11 x^{2} - 53 x - 33$ 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.

$f (x) = p = \pm 1, \pm 33, \pm 3, \pm 11$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 33, \pm 3, \pm 11$

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

Substitute $- 3$ into the polynomial.

$f (x) = {(- 3)}^{3} - 11 {(- 3)}^{2} - 53 \cdot - 3 - 33$

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

$f (x) = - 27 - 11 {(- 3)}^{2} - 53 \cdot - 3 - 33$

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

$f (x) = - 27 - 11 \cdot 9 - 53 \cdot - 3 - 33$

Multiply $- 11$ by $9$ .

$f (x) = - 27 - 99 - 53 \cdot - 3 - 33$

Subtract $99$ from $- 27$ .

$f (x) = - 126 - 53 \cdot - 3 - 33$

Multiply $- 53$ by $- 3$ .

$f (x) = - 126 + 159 - 33$

Add $- 126$ and $159$ .

$f (x) = 33 - 33$

Subtract $33$ from $33$ .

$f (x) = 0$

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

$f (x) = \frac{x^{3} - 11 x^{2} - 53 x - 33}{x + 3}$

Divide $x^{3} - 11 x^{2} - 53 x - 33$ by $x + 3$ .

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

$f (x) =$

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 $- 14 x^{2}$ 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 $- 14 x^{2} - 42 x$

$f (x) =$

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 $- 11 x$ 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 $- 11 x - 33$

$f (x) =$

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.

$f (x) = x^{2} - 14 x - 11$

Write $x^{3} - 11 x^{2} - 53 x - 33$ as a set of factors.

$f (x) = (x + 1) ((x + 3) (x^{2} - 14 x - 11))$

Remove unnecessary parentheses.

$f (x) = (x + 1) (x + 3) (x^{2} - 14 x - 11)$

Write as a Set of Linear Factors f(x)=x^4-10x^3-64x^2-86x-33

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