Write as a Set of Linear Factors p(x)=x^3-2x^2-5x+6

$p (x) = x^{3} - 2 x^{2} - 5 x + 6$

Step 1

Factor $x^{3} - 2 x^{2} - 5 x + 6$ using the rational roots test.

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.

$p (x) = p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

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

$p (x) = \pm 1, \pm 6, \pm 2, \pm 3$

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.

$p (x) = 1^{3} - 2 \cdot 1^{2} - 5 \cdot 1 + 6$

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

$p (x) = 1 - 2 \cdot 1^{2} - 5 \cdot 1 + 6$

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

$p (x) = 1 - 2 \cdot 1 - 5 \cdot 1 + 6$

Multiply $- 2$ by $1$ .

$p (x) = 1 - 2 - 5 \cdot 1 + 6$

Subtract $2$ from $1$ .

$p (x) = - 1 - 5 \cdot 1 + 6$

Multiply $- 5$ by $1$ .

$p (x) = - 1 - 5 + 6$

Subtract $5$ from $- 1$ .

$p (x) = - 6 + 6$

Add $- 6$ and $6$ .

$p (x) = 0$

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.

$p (x) = \frac{x^{3} - 2 x^{2} - 5 x + 6}{x - 1}$

Divide $x^{3} - 2 x^{2} - 5 x + 6$ 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 $0$ .

$p (x) =$

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

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

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

$p (x) =$

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

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

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

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 6$

$p (x) =$

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

$p (x) =$

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

$p (x) = x^{2} - x - 6$

Write $x^{3} - 2 x^{2} - 5 x + 6$ as a set of factors.

$p (x) = (x - 1) (x^{2} - x - 6)$

Step 2

Factor $x^{2} - x - 6$ using the AC method.

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$p (x) = - 3, 2$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Write as a Set of Linear Factors p(x)=x^3-2x^2-5x+6

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