# Algebra Examples

Step 1

Set equal to .

Step 2

Step 2.1

Factor the left side of the equation.

Step 2.1.1

Factor using the rational roots test.

Step 2.1.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.

Step 2.1.1.2

Find every combination of . These are the possible roots of the polynomial function.

Step 2.1.1.3

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

Step 2.1.1.3.1

Substitute into the polynomial.

Step 2.1.1.3.2

Raise to the power of .

Step 2.1.1.3.3

Raise to the power of .

Step 2.1.1.3.4

Multiply by .

Step 2.1.1.3.5

Subtract from .

Step 2.1.1.3.6

Multiply by .

Step 2.1.1.3.7

Subtract from .

Step 2.1.1.3.8

Add and .

Step 2.1.1.4

Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

Step 2.1.1.5

Divide by .

Step 2.1.1.5.1

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

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Step 2.1.1.5.2

Divide the highest order term in the dividend by the highest order term in divisor .

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Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

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Step 2.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

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Step 2.1.1.5.5

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

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Step 2.1.1.5.6

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

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Step 2.1.1.5.7

Divide the highest order term in the dividend by the highest order term in divisor .

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Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

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Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

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Step 2.1.1.5.10

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

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Step 2.1.1.5.11

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

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Step 2.1.1.5.12

Divide the highest order term in the dividend by the highest order term in divisor .

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Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

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Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

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Step 2.1.1.5.15

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

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Step 2.1.1.5.16

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

Step 2.1.1.6

Write as a set of factors.

Step 2.1.2

Factor using the AC method.

Step 2.1.2.1

Factor using the AC method.

Step 2.1.2.1.1

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Step 2.1.2.1.2

Write the factored form using these integers.

Step 2.1.2.2

Remove unnecessary parentheses.

Step 2.2

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Step 2.3

Set equal to and solve for .

Step 2.3.1

Set equal to .

Step 2.3.2

Add to both sides of the equation.

Step 2.4

Set equal to and solve for .

Step 2.4.1

Set equal to .

Step 2.4.2

Add to both sides of the equation.

Step 2.5

Set equal to and solve for .

Step 2.5.1

Set equal to .

Step 2.5.2

Subtract from both sides of the equation.

Step 2.6

The final solution is all the values that make true. The multiplicity of a root is the number of times the root appears.

(Multiplicity of )

(Multiplicity of )

(Multiplicity of )

(Multiplicity of )

(Multiplicity of )

(Multiplicity of )

Step 3