Algebra Examples

$w (x) = x^{4} + 5 x^{3} + 6 x^{2} - 3 x - 9$

Step 1

Regroup terms.

$w (x) = - 3 x - 9 + x^{4} + 5 x^{3} + 6 x^{2}$

Step 2

Factor $- 3$ out of $- 3 x - 9$ .

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Step 2.1

Factor $- 3$ out of $- 3 x$ .

$w (x) = - 3 x - 9 + x^{4} + 5 x^{3} + 6 x^{2}$

Step 2.2

Factor $- 3$ out of $- 9$ .

$w (x) = - 3 x - 3 \cdot 3 + x^{4} + 5 x^{3} + 6 x^{2}$

Step 2.3

Factor $- 3$ out of $- 3 (x) - 3 (3)$ .

$w (x) = - 3 (x + 3) + x^{4} + 5 x^{3} + 6 x^{2}$

Step 3

Factor $x^{2}$ out of $x^{4} + 5 x^{3} + 6 x^{2}$ .

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Step 3.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 3.2

Factor $x^{2}$ out of $5 x^{3}$ .

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

Step 3.3

Factor $x^{2}$ out of $6 x^{2}$ .

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

Step 3.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (5 x)$ .

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

Step 3.5

Factor $x^{2}$ out of $x^{2} (x^{2} + 5 x) + x^{2} \cdot 6$ .

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

Step 4

Factor.

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Step 4.1

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 4.1.1

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 $5$ .

$w (x) = 2, 3$

Step 4.1.2

Write the factored form using these integers.

$w (x) = - 3 (x + 3) + x^{2} ((x + 2) (x + 3))$

Step 4.2

Remove unnecessary parentheses.

$w (x) = - 3 (x + 3) + x^{2} (x + 2) (x + 3)$

Step 5

Factor $x + 3$ out of $- 3 (x + 3) + x^{2} (x + 2) (x + 3)$ .

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Step 5.1

Factor $x + 3$ out of $- 3 (x + 3)$ .

$w (x) = (x + 3) \cdot - 3 + x^{2} (x + 2) (x + 3)$

Step 5.2

Factor $x + 3$ out of $x^{2} (x + 2) (x + 3)$ .

$w (x) = (x + 3) \cdot - 3 + (x + 3) (x^{2} (x + 2))$

Step 5.3

Factor $x + 3$ out of $(x + 3) \cdot - 3 + (x + 3) (x^{2} (x + 2))$ .

$w (x) = (x + 3) (- 3 + x^{2} (x + 2))$

Step 6

Apply the distributive property.

$w (x) = (x + 3) (- 3 + x^{2} x + x^{2} \cdot 2)$

Step 7

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.1

Multiply $x^{2}$ by $x$ .

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Step 7.1.1

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

$w (x) = (x + 3) (- 3 + x^{2} x + x^{2} \cdot 2)$

Step 7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$w (x) = (x + 3) (- 3 + x^{2 + 1} + x^{2} \cdot 2)$

Step 7.2

Add $2$ and $1$ .

$w (x) = (x + 3) (- 3 + x^{3} + x^{2} \cdot 2)$

Step 8

Move $2$ to the left of $x^{2}$ .

$w (x) = (x + 3) (- 3 + x^{3} + 2 x^{2})$

Step 9

Reorder terms.

$w (x) = (x + 3) (x^{3} + 2 x^{2} - 3)$

Step 10

Factor.

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Step 10.1

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

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Step 10.1.1

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.

$w (x) = p = \pm 1, \pm 3$

$q = \pm 1$

Step 10.1.2

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

$w (x) = \pm 1, \pm 3$

Step 10.1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 10.1.3.1

Substitute $1$ into the polynomial.

$w (x) = 1^{3} + 2 \cdot 1^{2} - 3$

Step 10.1.3.2

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

$w (x) = 1 + 2 \cdot 1^{2} - 3$

Step 10.1.3.3

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

$w (x) = 1 + 2 \cdot 1 - 3$

Step 10.1.3.4

Multiply $2$ by $1$ .

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

Step 10.1.3.5

Add $1$ and $2$ .

$w (x) = 3 - 3$

Step 10.1.3.6

Subtract $3$ from $3$ .

$w (x) = 0$

Step 10.1.4

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.

$w (x) = \frac{x^{3} + 2 x^{2} - 3}{x - 1}$

Step 10.1.5

Divide $x^{3} + 2 x^{2} - 3$ by $x - 1$ .

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Step 10.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$w (x) =$

Step 10.1.5.2

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

$w (x) =$

Step 10.1.5.3

Multiply the new quotient term by the divisor.

$w (x) =$

Step 10.1.5.4

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

$w (x) =$

Step 10.1.5.5

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

$w (x) =$

Step 10.1.5.6

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

$w (x) =$

Step 10.1.5.7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

$w (x) =$

Step 10.1.5.8

Multiply the new quotient term by the divisor.

$w (x) =$

Step 10.1.5.9

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

$w (x) =$

Step 10.1.5.10

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

$w (x) =$

Step 10.1.5.11

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

$w (x) =$

Step 10.1.5.12

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

$w (x) =$

Step 10.1.5.13

Multiply the new quotient term by the divisor.

$w (x) =$

Step 10.1.5.14

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

$w (x) =$

Step 10.1.5.15

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

$w (x) =$

Step 10.1.5.16

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

$w (x) = x^{2} + 3 x + 3$

Step 10.1.6

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

$w (x) = (x + 3) ((x - 1) (x^{2} + 3 x + 3))$

Step 10.2

Remove unnecessary parentheses.

$w (x) = (x + 3) (x - 1) (x^{2} + 3 x + 3)$