Write as a Set of Linear Factors f(x)=x^4-10x^3-60x^2-70x-21

$f (x) = x^{4} - 10 x^{3} - 60 x^{2} - 70 x - 21$

Step 1

Factor $x^{4} - 10 x^{3} - 60 x^{2} - 70 x - 21$ 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.

$f (x) = p = \pm 1, \pm 21, \pm 3, \pm 7$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 21, \pm 3, \pm 7$

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.

$f (x) = {(- 1)}^{4} - 10 {(- 1)}^{3} - 60 {(- 1)}^{2} - 70 \cdot - 1 - 21$

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

$f (x) = 1 - 10 {(- 1)}^{3} - 60 {(- 1)}^{2} - 70 \cdot - 1 - 21$

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

$f (x) = 1 - 10 \cdot - 1 - 60 {(- 1)}^{2} - 70 \cdot - 1 - 21$

Multiply $- 10$ by $- 1$ .

$f (x) = 1 + 10 - 60 {(- 1)}^{2} - 70 \cdot - 1 - 21$

Add $1$ and $10$ .

$f (x) = 11 - 60 {(- 1)}^{2} - 70 \cdot - 1 - 21$

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

$f (x) = 11 - 60 \cdot 1 - 70 \cdot - 1 - 21$

Multiply $- 60$ by $1$ .

$f (x) = 11 - 60 - 70 \cdot - 1 - 21$

Subtract $60$ from $11$ .

$f (x) = - 49 - 70 \cdot - 1 - 21$

Multiply $- 70$ by $- 1$ .

$f (x) = - 49 + 70 - 21$

Add $- 49$ and $70$ .

$f (x) = 21 - 21$

Subtract $21$ from $21$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{4} - 10 x^{3} - 60 x^{2} - 70 x - 21}{x + 1}$

Divide $x^{4} - 10 x^{3} - 60 x^{2} - 70 x - 21$ 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 $- 49 x^{2}$ 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 $- 49 x^{2} - 49 x$

$f (x) =$

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 $- 21 x$ 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 $- 21 x - 21$

$f (x) =$

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.

$f (x) = x^{3} - 11 x^{2} - 49 x - 21$

Write $x^{4} - 10 x^{3} - 60 x^{2} - 70 x - 21$ as a set of factors.

$f (x) = (x + 1) (x^{3} - 11 x^{2} - 49 x - 21)$

Step 2

Factor $x^{3} - 11 x^{2} - 49 x - 21$ 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.

$f (x) = p = \pm 1, \pm 21, \pm 3, \pm 7$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 21, \pm 3, \pm 7$

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.

$f (x) = {(- 3)}^{3} - 11 {(- 3)}^{2} - 49 \cdot - 3 - 21$

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

$f (x) = - 27 - 11 {(- 3)}^{2} - 49 \cdot - 3 - 21$

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

$f (x) = - 27 - 11 \cdot 9 - 49 \cdot - 3 - 21$

Multiply $- 11$ by $9$ .

$f (x) = - 27 - 99 - 49 \cdot - 3 - 21$

Subtract $99$ from $- 27$ .

$f (x) = - 126 - 49 \cdot - 3 - 21$

Multiply $- 49$ by $- 3$ .

$f (x) = - 126 + 147 - 21$

Add $- 126$ and $147$ .

$f (x) = 21 - 21$

Subtract $21$ from $21$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} - 11 x^{2} - 49 x - 21}{x + 3}$

Divide $x^{3} - 11 x^{2} - 49 x - 21$ 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 $- 7 x$ 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 $- 7 x - 21$

$f (x) =$

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.

$f (x) = x^{2} - 14 x - 7$

Write $x^{3} - 11 x^{2} - 49 x - 21$ as a set of factors.

$f (x) = (x + 1) ((x + 3) (x^{2} - 14 x - 7))$

Remove unnecessary parentheses.

$f (x) = (x + 1) (x + 3) (x^{2} - 14 x - 7)$

Write as a Set of Linear Factors f(x)=x^4-10x^3-60x^2-70x-21

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