Write as a Set of Linear Factors h(x)=x^3-4x^2+14x-20

$h (x) = x^{3} - 4 x^{2} + 14 x - 20$

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.

$h (x) = p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

$q = \pm 1$

Step 2

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

$h (x) = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 3

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.

$h (x) = 2^{3} - 4 \cdot 2^{2} + 14 \cdot 2 - 20$

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

$h (x) = 8 - 4 \cdot 2^{2} + 14 \cdot 2 - 20$

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

$h (x) = 8 - 4 \cdot 4 + 14 \cdot 2 - 20$

Multiply $- 4$ by $4$ .

$h (x) = 8 - 16 + 14 \cdot 2 - 20$

Subtract $16$ from $8$ .

$h (x) = - 8 + 14 \cdot 2 - 20$

Multiply $14$ by $2$ .

$h (x) = - 8 + 28 - 20$

Add $- 8$ and $28$ .

$h (x) = 20 - 20$

Subtract $20$ from $20$ .

$h (x) = 0$

Step 4

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.

$h (x) = \frac{x^{3} - 4 x^{2} + 14 x - 20}{x - 2}$

Step 5

Divide $x^{3} - 4 x^{2} + 14 x - 20$ 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 $0$ .

$h (x) =$

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

$h (x) =$

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

$h (x) =$

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 $10 x$ 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 $10 x - 20$

$h (x) =$

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.

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

Step 6

Write $x^{3} - 4 x^{2} + 14 x - 20$ as a set of factors.

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

Write as a Set of Linear Factors h(x)=x^3-4x^2+14x-20

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