Write as a Set of Linear Factors f(x)=x^4-2x^3+2x-1

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

Step 1

Factor $x^{4} - 2 x^{3} + 2 x - 1$ 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.

$f (x) = p = \pm 1$

$q = \pm 1$

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

$f (x) = \pm 1$

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.

$f (x) = {(- 1)}^{4} - 2 {(- 1)}^{3} + 2 \cdot - 1 - 1$

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

$f (x) = 1 - 2 {(- 1)}^{3} + 2 \cdot - 1 - 1$

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

$f (x) = 1 - 2 \cdot - 1 + 2 \cdot - 1 - 1$

Multiply $- 2$ by $- 1$ .

$f (x) = 1 + 2 + 2 \cdot - 1 - 1$

Add $1$ and $2$ .

$f (x) = 3 + 2 \cdot - 1 - 1$

Multiply $2$ by $- 1$ .

$f (x) = 3 - 2 - 1$

Subtract $2$ from $3$ .

$f (x) = 1 - 1$

Subtract $1$ from $1$ .

$f (x) = 0$

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.

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

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

$f (x) =$

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

$f (x) =$

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 $- x$ 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 $- x - 1$

$f (x) =$

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.

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

Write $x^{4} - 2 x^{3} + 2 x - 1$ as a set of factors.

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

Step 2

Factor $x^{3} - 3 x^{2} + 3 x - 1$ 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.

$f (x) = p = \pm 1$

$q = \pm 1$

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

$f (x) = \pm 1$

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.

$f (x) = 1^{3} - 3 \cdot 1^{2} + 3 \cdot 1 - 1$

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

$f (x) = 1 - 3 \cdot 1^{2} + 3 \cdot 1 - 1$

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

$f (x) = 1 - 3 \cdot 1 + 3 \cdot 1 - 1$

Multiply $- 3$ by $1$ .

$f (x) = 1 - 3 + 3 \cdot 1 - 1$

Subtract $3$ from $1$ .

$f (x) = - 2 + 3 \cdot 1 - 1$

Multiply $3$ by $1$ .

$f (x) = - 2 + 3 - 1$

Add $- 2$ and $3$ .

$f (x) = 1 - 1$

Subtract $1$ from $1$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} - 3 x^{2} + 3 x - 1}{x - 1}$

Divide $x^{3} - 3 x^{2} + 3 x - 1$ 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 $x^{3}$ 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 $x^{3} - x^{2}$

$f (x) =$

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

$f (x) =$

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 $x$ 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 $x - 1$

$f (x) =$

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.

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

Write $x^{3} - 3 x^{2} + 3 x - 1$ as a set of factors.

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

Step 3

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$f (x) = (x + 1) ((x - 1) (x^{2} - 2 \cdot x \cdot 1 + 1^{2}))$

Factor using the perfect square trinomial rule $a^{2} - 2 a i + i^{2} = {(a - i)}^{2}$ , where $a = x$ and $i = 1$ .

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

Step 4

Combine like factors.

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

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

Add $1$ and $2$ .

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

Write as a Set of Linear Factors f(x)=x^4-2x^3+2x-1

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