Write as a Set of Linear Factors f(x)=x^3+2x^2+5x-26

$f (x) = x^{3} + 2 x^{2} + 5 x - 26$

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.

$f (x) = p = \pm 1, \pm 26, \pm 2, \pm 13$

$q = \pm 1$

Step 2

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

$f (x) = \pm 1, \pm 26, \pm 2, \pm 13$

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.

$f (x) = 2^{3} + 2 \cdot 2^{2} + 5 \cdot 2 - 26$

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

$f (x) = 8 + 2 \cdot 2^{2} + 5 \cdot 2 - 26$

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

$f (x) = 8 + 2 \cdot 4 + 5 \cdot 2 - 26$

Multiply $2$ by $4$ .

$f (x) = 8 + 8 + 5 \cdot 2 - 26$

Add $8$ and $8$ .

$f (x) = 16 + 5 \cdot 2 - 26$

Multiply $5$ by $2$ .

$f (x) = 16 + 10 - 26$

Add $16$ and $10$ .

$f (x) = 26 - 26$

Subtract $26$ from $26$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} + 2 x^{2} + 5 x - 26}{x - 2}$

Step 5

Divide $x^{3} + 2 x^{2} + 5 x - 26$ 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 $0$ .

$f (x) =$

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

$f (x) =$

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 $4 x^{2}$ 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 $4 x^{2} - 8 x$

$f (x) =$

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 $13 x$ 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 $13 x - 26$

$f (x) =$

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.

$f (x) = x^{2} + 4 x + 13$

Step 6

Write $x^{3} + 2 x^{2} + 5 x - 26$ as a set of factors.

$f (x) = (x - 2) (x^{2} + 4 x + 13)$

Write as a Set of Linear Factors f(x)=x^3+2x^2+5x-26

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