Write as a Set of Linear Factors f(x)=x^4-11x^3-53x^2-67x-26

$f (x) = x^{4} - 11 x^{3} - 53 x^{2} - 67 x - 26$

Step 1

Factor $x^{4} - 11 x^{3} - 53 x^{2} - 67 x - 26$ 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.

$f (x) = p = \pm 1, \pm 26, \pm 2, \pm 13$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 26, \pm 2, \pm 13$

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.

$f (x) = {(- 1)}^{4} - 11 {(- 1)}^{3} - 53 {(- 1)}^{2} - 67 \cdot - 1 - 26$

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

$f (x) = 1 - 11 {(- 1)}^{3} - 53 {(- 1)}^{2} - 67 \cdot - 1 - 26$

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

$f (x) = 1 - 11 \cdot - 1 - 53 {(- 1)}^{2} - 67 \cdot - 1 - 26$

Multiply $- 11$ by $- 1$ .

$f (x) = 1 + 11 - 53 {(- 1)}^{2} - 67 \cdot - 1 - 26$

Add $1$ and $11$ .

$f (x) = 12 - 53 {(- 1)}^{2} - 67 \cdot - 1 - 26$

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

$f (x) = 12 - 53 \cdot 1 - 67 \cdot - 1 - 26$

Multiply $- 53$ by $1$ .

$f (x) = 12 - 53 - 67 \cdot - 1 - 26$

Subtract $53$ from $12$ .

$f (x) = - 41 - 67 \cdot - 1 - 26$

Multiply $- 67$ by $- 1$ .

$f (x) = - 41 + 67 - 26$

Add $- 41$ and $67$ .

$f (x) = 26 - 26$

Subtract $26$ from $26$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{4} - 11 x^{3} - 53 x^{2} - 67 x - 26}{x + 1}$

Divide $x^{4} - 11 x^{3} - 53 x^{2} - 67 x - 26$ 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 $- 12 x^{3}$ 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 $- 12 x^{3} - 12 x^{2}$

$f (x) =$

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 $- 41 x^{2}$ 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 $- 41 x^{2} - 41 x$

$f (x) =$

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 $- 26 x$ 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 $- 26 x - 26$

$f (x) =$

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.

$f (x) = x^{3} - 12 x^{2} - 41 x - 26$

Write $x^{4} - 11 x^{3} - 53 x^{2} - 67 x - 26$ as a set of factors.

$f (x) = (x + 1) (x^{3} - 12 x^{2} - 41 x - 26)$

Step 2

Factor $x^{3} - 12 x^{2} - 41 x - 26$ 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.

$f (x) = p = \pm 1, \pm 26, \pm 2, \pm 13$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 26, \pm 2, \pm 13$

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.

$f (x) = {(- 2)}^{3} - 12 {(- 2)}^{2} - 41 \cdot - 2 - 26$

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

$f (x) = - 8 - 12 {(- 2)}^{2} - 41 \cdot - 2 - 26$

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

$f (x) = - 8 - 12 \cdot 4 - 41 \cdot - 2 - 26$

Multiply $- 12$ by $4$ .

$f (x) = - 8 - 48 - 41 \cdot - 2 - 26$

Subtract $48$ from $- 8$ .

$f (x) = - 56 - 41 \cdot - 2 - 26$

Multiply $- 41$ by $- 2$ .

$f (x) = - 56 + 82 - 26$

Add $- 56$ and $82$ .

$f (x) = 26 - 26$

Subtract $26$ from $26$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} - 12 x^{2} - 41 x - 26}{x + 2}$

Divide $x^{3} - 12 x^{2} - 41 x - 26$ 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 $- 14 x^{2}$ 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 $- 14 x^{2} - 28 x$

$f (x) =$

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 $- 13 x$ 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 $- 13 x - 26$

$f (x) =$

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.

$f (x) = x^{2} - 14 x - 13$

Write $x^{3} - 12 x^{2} - 41 x - 26$ as a set of factors.

$f (x) = (x + 1) ((x + 2) (x^{2} - 14 x - 13))$

Remove unnecessary parentheses.

$f (x) = (x + 1) (x + 2) (x^{2} - 14 x - 13)$

Write as a Set of Linear Factors f(x)=x^4-11x^3-53x^2-67x-26

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