Write as a Set of Linear Factors P(x)=x^5-8x^4+3x^3+82x^2-78x

$P (x) = x^{5} - 8 x^{4} + 3 x^{3} + 82 x^{2} - 78 x$

Step 1

Factor $x$ out of $x^{5} - 8 x^{4} + 3 x^{3} + 82 x^{2} - 78 x$ .

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

$P (x) = x \cdot x^{4} - 8 x^{4} + 3 x^{3} + 82 x^{2} - 78 x$

Factor $x$ out of $- 8 x^{4}$ .

$P (x) = x \cdot x^{4} + x (- 8 x^{3}) + 3 x^{3} + 82 x^{2} - 78 x$

Factor $x$ out of $3 x^{3}$ .

$P (x) = x \cdot x^{4} + x (- 8 x^{3}) + x (3 x^{2}) + 82 x^{2} - 78 x$

Factor $x$ out of $82 x^{2}$ .

$P (x) = x \cdot x^{4} + x (- 8 x^{3}) + x (3 x^{2}) + x (82 x) - 78 x$

Factor $x$ out of $- 78 x$ .

$P (x) = x \cdot x^{4} + x (- 8 x^{3}) + x (3 x^{2}) + x (82 x) + x \cdot - 78$

Factor $x$ out of $x \cdot x^{4} + x (- 8 x^{3})$ .

$P (x) = x (x^{4} - 8 x^{3}) + x (3 x^{2}) + x (82 x) + x \cdot - 78$

Factor $x$ out of $x (x^{4} - 8 x^{3}) + x (3 x^{2})$ .

$P (x) = x (x^{4} - 8 x^{3} + 3 x^{2}) + x (82 x) + x \cdot - 78$

Factor $x$ out of $x (x^{4} - 8 x^{3} + 3 x^{2}) + x (82 x)$ .

$P (x) = x (x^{4} - 8 x^{3} + 3 x^{2} + 82 x) + x \cdot - 78$

Factor $x$ out of $x (x^{4} - 8 x^{3} + 3 x^{2} + 82 x) + x \cdot - 78$ .

$P (x) = x (x^{4} - 8 x^{3} + 3 x^{2} + 82 x - 78)$

Step 2

Factor over the complex numbers.

Factor $x^{4} - 8 x^{3} + 3 x^{2} + 82 x - 78$ 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.

$P (x) = p = \pm 1, \pm 78, \pm 2, \pm 39, \pm 3, \pm 26, \pm 6, \pm 13$

$q = \pm 1$

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

$P (x) = \pm 1, \pm 78, \pm 2, \pm 39, \pm 3, \pm 26, \pm 6, \pm 13$

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.

$P (x) = 1^{4} - 8 \cdot 1^{3} + 3 \cdot 1^{2} + 82 \cdot 1 - 78$

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

$P (x) = 1 - 8 \cdot 1^{3} + 3 \cdot 1^{2} + 82 \cdot 1 - 78$

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

$P (x) = 1 - 8 \cdot 1 + 3 \cdot 1^{2} + 82 \cdot 1 - 78$

Multiply $- 8$ by $1$ .

$P (x) = 1 - 8 + 3 \cdot 1^{2} + 82 \cdot 1 - 78$

Subtract $8$ from $1$ .

$P (x) = - 7 + 3 \cdot 1^{2} + 82 \cdot 1 - 78$

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

$P (x) = - 7 + 3 \cdot 1 + 82 \cdot 1 - 78$

Multiply $3$ by $1$ .

$P (x) = - 7 + 3 + 82 \cdot 1 - 78$

Add $- 7$ and $3$ .

$P (x) = - 4 + 82 \cdot 1 - 78$

Multiply $82$ by $1$ .

$P (x) = - 4 + 82 - 78$

Add $- 4$ and $82$ .

$P (x) = 78 - 78$

Subtract $78$ from $78$ .

$P (x) = 0$

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.

$P (x) = \frac{x^{4} - 8 x^{3} + 3 x^{2} + 82 x - 78}{x - 1}$

Divide $x^{4} - 8 x^{3} + 3 x^{2} + 82 x - 78$ 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 $0$ .

$P (x) =$

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

$P (x) =$

Multiply the new quotient term by the divisor.

$P (x) =$

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

$P (x) =$

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

$P (x) =$

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

$P (x) =$

Divide the highest order term in the dividend $- 7 x^{3}$ by the highest order term in divisor $x$ .

$P (x) =$

Multiply the new quotient term by the divisor.

$P (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{3} + 7 x^{2}$

$P (x) =$

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

$P (x) =$

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

$P (x) =$

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

$P (x) =$

Multiply the new quotient term by the divisor.

$P (x) =$

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

$P (x) =$

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

$P (x) =$

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

$P (x) =$

Divide the highest order term in the dividend $78 x$ by the highest order term in divisor $x$ .

$P (x) =$

Multiply the new quotient term by the divisor.

$P (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $78 x - 78$

$P (x) =$

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

$P (x) =$

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

$P (x) = x^{3} - 7 x^{2} - 4 x + 78$

Write $x^{4} - 8 x^{3} + 3 x^{2} + 82 x - 78$ as a set of factors.

$P (x) = x ((x - 1) (x^{3} - 7 x^{2} - 4 x + 78))$

Factor.

Factor $x^{3} - 7 x^{2} - 4 x + 78$ 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.

$P (x) = p = \pm 1, \pm 78, \pm 2, \pm 39, \pm 3, \pm 26, \pm 6, \pm 13$

$q = \pm 1$

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

$P (x) = \pm 1, \pm 78, \pm 2, \pm 39, \pm 3, \pm 26, \pm 6, \pm 13$

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.

$P (x) = {(- 3)}^{3} - 7 {(- 3)}^{2} - 4 \cdot - 3 + 78$

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

$P (x) = - 27 - 7 {(- 3)}^{2} - 4 \cdot - 3 + 78$

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

$P (x) = - 27 - 7 \cdot 9 - 4 \cdot - 3 + 78$

Multiply $- 7$ by $9$ .

$P (x) = - 27 - 63 - 4 \cdot - 3 + 78$

Subtract $63$ from $- 27$ .

$P (x) = - 90 - 4 \cdot - 3 + 78$

Multiply $- 4$ by $- 3$ .

$P (x) = - 90 + 12 + 78$

Add $- 90$ and $12$ .

$P (x) = - 78 + 78$

Add $- 78$ and $78$ .

$P (x) = 0$

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.

$P (x) = \frac{x^{3} - 7 x^{2} - 4 x + 78}{x + 3}$

Divide $x^{3} - 7 x^{2} - 4 x + 78$ 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 $0$ .

$P (x) =$

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

$P (x) =$

Multiply the new quotient term by the divisor.

$P (x) =$

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

$P (x) =$

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

$P (x) =$

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

$P (x) =$

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

$P (x) =$

Multiply the new quotient term by the divisor.

$P (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 30 x$

$P (x) =$

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

$P (x) =$

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

$P (x) =$

Divide the highest order term in the dividend $26 x$ by the highest order term in divisor $x$ .

$P (x) =$

Multiply the new quotient term by the divisor.

$P (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $26 x + 78$

$P (x) =$

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

$P (x) =$

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

$P (x) = x^{2} - 10 x + 26$

Write $x^{3} - 7 x^{2} - 4 x + 78$ as a set of factors.

$P (x) = x ((x - 1) ((x + 3) (x^{2} - 10 x + 26)))$

Remove unnecessary parentheses.

$P (x) = x ((x - 1) (x + 3) (x^{2} - 10 x + 26))$

Remove unnecessary parentheses.

$P (x) = x (x - 1) (x + 3) (x^{2} - 10 x + 26)$

Write as a Set of Linear Factors P(x)=x^5-8x^4+3x^3+82x^2-78x

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