Write as a Set of Linear Factors h(x)=x^3+7x^2+10x+4

$h (x) = x^{3} + 7 x^{2} + 10 x + 4$

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.

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

$q = \pm 1$

Step 2

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

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

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.

$h (x) = {(- 1)}^{3} + 7 {(- 1)}^{2} + 10 \cdot - 1 + 4$

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

$h (x) = - 1 + 7 {(- 1)}^{2} + 10 \cdot - 1 + 4$

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

$h (x) = - 1 + 7 \cdot 1 + 10 \cdot - 1 + 4$

Multiply $7$ by $1$ .

$h (x) = - 1 + 7 + 10 \cdot - 1 + 4$

Add $- 1$ and $7$ .

$h (x) = 6 + 10 \cdot - 1 + 4$

Multiply $10$ by $- 1$ .

$h (x) = 6 - 10 + 4$

Subtract $10$ from $6$ .

$h (x) = - 4 + 4$

Add $- 4$ and $4$ .

$h (x) = 0$

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.

$h (x) = \frac{x^{3} + 7 x^{2} + 10 x + 4}{x + 1}$

Step 5

Divide $x^{3} + 7 x^{2} + 10 x + 4$ 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 $6 x^{2}$ 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 $6 x^{2} + 6 x$

$h (x) =$

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 $4 x$ 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 $4 x + 4$

$h (x) =$

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.

$h (x) = x^{2} + 6 x + 4$

Step 6

Write $x^{3} + 7 x^{2} + 10 x + 4$ as a set of factors.

$h (x) = (x + 1) (x^{2} + 6 x + 4)$

Write as a Set of Linear Factors h(x)=x^3+7x^2+10x+4

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