Write as a Set of Linear Factors h(x)=x^3-5x^2+9x-5

$h (x) = x^{3} - 5 x^{2} + 9 x - 5$

Step 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.

$h (x) = p = \pm 1, \pm 5$

$q = \pm 1$

Step 2

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

$h (x) = \pm 1, \pm 5$

Step 3

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.

$h (x) = 1^{3} - 5 \cdot 1^{2} + 9 \cdot 1 - 5$

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

$h (x) = 1 - 5 \cdot 1^{2} + 9 \cdot 1 - 5$

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

$h (x) = 1 - 5 \cdot 1 + 9 \cdot 1 - 5$

Multiply $- 5$ by $1$ .

$h (x) = 1 - 5 + 9 \cdot 1 - 5$

Subtract $5$ from $1$ .

$h (x) = - 4 + 9 \cdot 1 - 5$

Multiply $9$ by $1$ .

$h (x) = - 4 + 9 - 5$

Add $- 4$ and $9$ .

$h (x) = 5 - 5$

Subtract $5$ from $5$ .

$h (x) = 0$

Step 4

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.

$h (x) = \frac{x^{3} - 5 x^{2} + 9 x - 5}{x - 1}$

Step 5

Divide $x^{3} - 5 x^{2} + 9 x - 5$ 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 $0$ .

$h (x) =$

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

$h (x) =$

Multiply the new quotient term by the divisor.

$h (x) =$

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

$h (x) =$

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

$h (x) =$

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

$h (x) =$

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

$h (x) =$

Multiply the new quotient term by the divisor.

$h (x) =$

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

$h (x) =$

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

$h (x) =$

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

$h (x) =$

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $x$ .

$h (x) =$

Multiply the new quotient term by the divisor.

$h (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x - 5$

$h (x) =$

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

$h (x) =$

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

$h (x) = x^{2} - 4 x + 5$

Step 6

Write $x^{3} - 5 x^{2} + 9 x - 5$ as a set of factors.

$h (x) = (x - 1) (x^{2} - 4 x + 5)$

Write as a Set of Linear Factors h(x)=x^3-5x^2+9x-5

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