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

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

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.

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

$q = \pm 1$

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

$g (x) = \pm 1$

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.

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $- 3$ by $1$ .

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

Subtract $3$ from $2$ .

$g (x) = - 1 + 1$

Add $- 1$ and $1$ .

$g (x) = 0$

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.

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

Divide $x^{3} + x^{2} - 3 x + 1$ 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 $0$ .

$g (x) =$

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

$g (x) =$

Multiply the new quotient term by the divisor.

$g (x) =$

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

$g (x) =$

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

$g (x) =$

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

$g (x) =$

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

$g (x) =$

Multiply the new quotient term by the divisor.

$g (x) =$

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

$g (x) =$

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

$g (x) =$

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

$g (x) =$

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

$g (x) =$

Multiply the new quotient term by the divisor.

$g (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- x + 1$

$g (x) =$

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

$g (x) =$

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

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

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

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

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

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