Calculus Examples

Factor x^3-3x-18
Step 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
Find every combination of . These are the possible roots of the polynomial function.
Step 3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Step 3.1
Substitute into the polynomial.
Step 3.2
Raise to the power of .
Step 3.3
Multiply by .
Step 3.4
Subtract from .
Step 3.5
Subtract from .
Step 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 5
Divide by .
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Step 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 5.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.3
Multiply the new quotient term by the divisor.
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Step 5.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.6
Pull the next terms from the original dividend down into the current dividend.
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Step 5.7
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.8
Multiply the new quotient term by the divisor.
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Step 5.9
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.11
Pull the next terms from the original dividend down into the current dividend.
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Step 5.12
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.13
Multiply the new quotient term by the divisor.
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Step 5.14
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.16
Since the remander is , the final answer is the quotient.
Step 6
Write as a set of factors.