Write as a Set of Linear Factors f(x)=x^4+3x^3-30x^2-124x-120

$f (x) = x^{4} + 3 x^{3} - 30 x^{2} - 124 x - 120$

Step 1

Factor $x^{4} + 3 x^{3} - 30 x^{2} - 124 x - 120$ 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.

$f (x) = p = \pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

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.

$f (x) = {(- 2)}^{4} + 3 {(- 2)}^{3} - 30 {(- 2)}^{2} - 124 \cdot - 2 - 120$

Raise $- 2$ to the power of $4$ .

$f (x) = 16 + 3 {(- 2)}^{3} - 30 {(- 2)}^{2} - 124 \cdot - 2 - 120$

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

$f (x) = 16 + 3 \cdot - 8 - 30 {(- 2)}^{2} - 124 \cdot - 2 - 120$

Multiply $3$ by $- 8$ .

$f (x) = 16 - 24 - 30 {(- 2)}^{2} - 124 \cdot - 2 - 120$

Subtract $24$ from $16$ .

$f (x) = - 8 - 30 {(- 2)}^{2} - 124 \cdot - 2 - 120$

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

$f (x) = - 8 - 30 \cdot 4 - 124 \cdot - 2 - 120$

Multiply $- 30$ by $4$ .

$f (x) = - 8 - 120 - 124 \cdot - 2 - 120$

Subtract $120$ from $- 8$ .

$f (x) = - 128 - 124 \cdot - 2 - 120$

Multiply $- 124$ by $- 2$ .

$f (x) = - 128 + 248 - 120$

Add $- 128$ and $248$ .

$f (x) = 120 - 120$

Subtract $120$ from $120$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{4} + 3 x^{3} - 30 x^{2} - 124 x - 120}{x + 2}$

Divide $x^{4} + 3 x^{3} - 30 x^{2} - 124 x - 120$ 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 $x^{4}$ 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 $x^{4} + 2 x^{3}$

$f (x) =$

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

$f (x) =$

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 $- 32 x^{2}$ 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 $- 32 x^{2} - 64 x$

$f (x) =$

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 $- 60 x$ 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 $- 60 x - 120$

$f (x) =$

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.

$f (x) = x^{3} + x^{2} - 32 x - 60$

Write $x^{4} + 3 x^{3} - 30 x^{2} - 124 x - 120$ as a set of factors.

$f (x) = (x + 2) (x^{3} + x^{2} - 32 x - 60)$

Step 2

Factor $x^{3} + x^{2} - 32 x - 60$ 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.

$f (x) = p = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

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.

$f (x) = {(- 2)}^{3} + {(- 2)}^{2} - 32 \cdot - 2 - 60$

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

$f (x) = - 8 + {(- 2)}^{2} - 32 \cdot - 2 - 60$

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

$f (x) = - 8 + 4 - 32 \cdot - 2 - 60$

Add $- 8$ and $4$ .

$f (x) = - 4 - 32 \cdot - 2 - 60$

Multiply $- 32$ by $- 2$ .

$f (x) = - 4 + 64 - 60$

Add $- 4$ and $64$ .

$f (x) = 60 - 60$

Subtract $60$ from $60$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} + x^{2} - 32 x - 60}{x + 2}$

Divide $x^{3} + x^{2} - 32 x - 60$ 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 $- 30 x$ 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 $- 30 x - 60$

$f (x) =$

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.

$f (x) = x^{2} - x - 30$

Write $x^{3} + x^{2} - 32 x - 60$ as a set of factors.

$f (x) = (x + 2) ((x + 2) (x^{2} - x - 30))$

Step 3

Factor $x^{2} - x - 30$ using the AC method.

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $- 30$ and whose sum is $- 1$ .

$f (x) = - 6, 5$

Write the factored form using these integers.

$f (x) = (x + 2) ((x + 2) ((x - 6) (x + 5)))$

Remove unnecessary parentheses.

$f (x) = (x + 2) ((x + 2) (x - 6) (x + 5))$

Remove unnecessary parentheses.

$f (x) = (x + 2) (x + 2) (x - 6) (x + 5)$

Step 4

Combine like factors.

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

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

Add $1$ and $1$ .

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

Write as a Set of Linear Factors f(x)=x^4+3x^3-30x^2-124x-120

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