Write as a Set of Linear Factors P(x)=2x^4-x^3-4x^2+10x-4

$P (x) = 2 x^{4} - x^{3} - 4 x^{2} + 10 x - 4$

Step 1

Factor $2 x^{4} - x^{3} - 4 x^{2} + 10 x - 4$ 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.

$P (x) = p = \pm 1, \pm 4, \pm 2$

$q = \pm 1, \pm 2$

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

$P (x) = \pm 1, \pm 0.5, \pm 4, \pm 2$

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Substitute $0.5$ into the polynomial.

$P (x) = 2 \cdot {0.5}^{4} - {0.5}^{3} - 4 \cdot {0.5}^{2} + 10 \cdot 0.5 - 4$

Raise $0.5$ to the power of $4$ .

$P (x) = 2 \cdot 0.0625 - {0.5}^{3} - 4 \cdot {0.5}^{2} + 10 \cdot 0.5 - 4$

Multiply $2$ by $0.0625$ .

$P (x) = 0.125 - {0.5}^{3} - 4 \cdot {0.5}^{2} + 10 \cdot 0.5 - 4$

Raise $0.5$ to the power of $3$ .

$P (x) = 0.125 - 1 \cdot 0.125 - 4 \cdot {0.5}^{2} + 10 \cdot 0.5 - 4$

Multiply $- 1$ by $0.125$ .

$P (x) = 0.125 - 0.125 - 4 \cdot {0.5}^{2} + 10 \cdot 0.5 - 4$

Subtract $0.125$ from $0.125$ .

$P (x) = 0 - 4 \cdot {0.5}^{2} + 10 \cdot 0.5 - 4$

Raise $0.5$ to the power of $2$ .

$P (x) = 0 - 4 \cdot 0.25 + 10 \cdot 0.5 - 4$

Multiply $- 4$ by $0.25$ .

$P (x) = 0 - 1 + 10 \cdot 0.5 - 4$

Subtract $1$ from $0$ .

$P (x) = - 1 + 10 \cdot 0.5 - 4$

Multiply $10$ by $0.5$ .

$P (x) = - 1 + 5 - 4$

Add $- 1$ and $5$ .

$P (x) = 4 - 4$

Subtract $4$ from $4$ .

$P (x) = 0$

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$P (x) = \frac{2 x^{4} - x^{3} - 4 x^{2} + 10 x - 4}{2 x - 1}$

Divide $2 x^{4} - x^{3} - 4 x^{2} + 10 x - 4$ by $2 x - 1$ .

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

$P (x) =$

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 $2 x^{4} - x^{3}$

$P (x) =$

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

$P (x) =$

Pull the next term 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 $2 x$ .

$P (x) =$

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 $- 4 x^{2} + 2 x$

$P (x) =$

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 $8 x$ by the highest order term in divisor $2 x$ .

$P (x) =$

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 $8 x - 4$

$P (x) =$

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.

$P (x) = x^{3} + 0 x^{2} - 2 x + 4$

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

$P (x) = (2 x - 1) (x^{3} + 0 x^{2} - 2 x + 4)$

Step 2

Multiply $0$ by $x^{2}$ .

$P (x) = (2 x - 1) (x^{3} + 0 - 2 x + 4)$

Step 3

Add $x^{3}$ and $0$ .

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

Step 4

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

$P (x) = p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

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

$P (x) = \pm 1, \pm 4, \pm 2$

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.

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

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

$P (x) = - 8 - 2 \cdot - 2 + 4$

Multiply $- 2$ by $- 2$ .

$P (x) = - 8 + 4 + 4$

Add $- 8$ and $4$ .

$P (x) = - 4 + 4$

Add $- 4$ and $4$ .

$P (x) = 0$

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.

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

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

$P (x) =$

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

$P (x) =$

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 $- 2 x^{2}$ 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 $- 2 x^{2} - 4 x$

$P (x) =$

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 $2 x$ 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 $2 x + 4$

$P (x) =$

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.

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

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

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

Remove unnecessary parentheses.

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

Write as a Set of Linear Factors P(x)=2x^4-x^3-4x^2+10x-4

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