# Algebra Examples

Step 1

Step 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 1.2

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

Step 1.3

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

Step 1.3.1

Substitute into the polynomial.

Step 1.3.2

Raise to the power of .

Step 1.3.3

Raise to the power of .

Step 1.3.4

Multiply by .

Step 1.3.5

Subtract from .

Step 1.3.6

Multiply by .

Step 1.3.7

Add and .

Step 1.3.8

Subtract from .

Step 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 1.5

Divide by .

Step 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 1.5.2

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

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Step 1.5.3

Multiply the new quotient term by the divisor.

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Step 1.5.4

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

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Step 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 1.5.6

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

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Step 1.5.7

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

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Step 1.5.8

Multiply the new quotient term by the divisor.

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Step 1.5.9

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

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Step 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 1.5.11

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

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Step 1.5.12

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

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Step 1.5.13

Multiply the new quotient term by the divisor.

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Step 1.5.14

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

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Step 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 1.5.16

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

Step 1.6

Write as a set of factors.

Step 2

Step 2.1

Rewrite as .

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

Step 2.3

Rewrite the polynomial.

Step 2.4

Factor using the perfect square trinomial rule , where and .

Step 3

Step 3.1

Raise to the power of .

Step 3.2

Use the power rule to combine exponents.

Step 3.3

Add and .

Step 4

Since the polynomial can be factored, it is not prime.

Not prime