# Algebra Examples

Step 1

Step 1.1

Factor using the rational roots test.

Step 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 1.1.2

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

Step 1.1.3

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

Step 1.1.3.1

Substitute into the polynomial.

Step 1.1.3.2

Raise to the power of .

Step 1.1.3.3

Raise to the power of .

Step 1.1.3.4

Multiply by .

Step 1.1.3.5

Add and .

Step 1.1.3.6

Add and .

Step 1.1.3.7

Subtract from .

Step 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 1.1.5

Divide by .

Step 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 1.1.5.2

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

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Step 1.1.5.3

Multiply the new quotient term by the divisor.

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Step 1.1.5.4

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

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Step 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 1.1.5.6

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

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Step 1.1.5.7

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

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Step 1.1.5.8

Multiply the new quotient term by the divisor.

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Step 1.1.5.9

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

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Step 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 1.1.5.11

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

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Step 1.1.5.12

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

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Step 1.1.5.13

Multiply the new quotient term by the divisor.

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Step 1.1.5.14

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

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Step 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 1.1.5.16

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

Step 1.1.6

Write as a set of factors.

Step 1.2

Factor using the AC method.

Step 1.2.1

Factor using the AC method.

Step 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 1.2.1.2

Write the factored form using these integers.

Step 1.2.2

Remove unnecessary parentheses.

Step 2

Step 2.1

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 2.2

Write the factored form using these integers.

Step 3

Step 3.1

Cancel the common factor.

Step 3.2

Rewrite the expression.

Step 4

Step 4.1

Cancel the common factor.

Step 4.2

Divide by .

Step 5

To find the holes in the graph, look at the denominator factors that were cancelled.

Step 6

Step 6.1

Set equal to .

Step 6.2

Subtract from both sides of the equation.

Step 6.3

Substitute for in and simplify.

Step 6.3.1

Substitute for to find the coordinate of the hole.

Step 6.3.2

Subtract from .

Step 6.4

Set equal to .

Step 6.5

Subtract from both sides of the equation.

Step 6.6

Substitute for in and simplify.

Step 6.6.1

Substitute for to find the coordinate of the hole.

Step 6.6.2

Subtract from .

Step 6.7

The holes in the graph are the points where any of the cancelled factors are equal to .

Step 7