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Precalculus Examples
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 the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is , which means it is a root.
Step 4
Step 4.1
Simplify each term.
Step 4.1.1
One to any power is one.
Step 4.1.2
One to any power is one.
Step 4.1.3
Multiply by .
Step 4.1.4
Multiply by .
Step 4.2
Simplify by adding and subtracting.
Step 4.2.1
Subtract from .
Step 4.2.2
Subtract from .
Step 4.2.3
Add and .
Step 5
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 6
Step 6.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
Step 6.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
Step 6.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 6.4
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
Step 6.5
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 6.6
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
Step 6.7
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 6.8
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
Step 6.9
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 6.10
Simplify the quotient polynomial.
Step 7
Step 7.1
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 7.2
Write the factored form using these integers.
Step 8
Step 8.1
Factor using the rational roots test.
Step 8.1.1
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 8.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 8.1.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
Step 8.1.3.1
Substitute into the polynomial.
Step 8.1.3.2
Raise to the power of .
Step 8.1.3.3
Raise to the power of .
Step 8.1.3.4
Multiply by .
Step 8.1.3.5
Subtract from .
Step 8.1.3.6
Multiply by .
Step 8.1.3.7
Subtract from .
Step 8.1.3.8
Add and .
Step 8.1.4
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 8.1.5
Divide by .
Step 8.1.5.1
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 8.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 8.1.5.3
Multiply the new quotient term by the divisor.
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Step 8.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 8.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 8.1.5.6
Pull the next terms from the original dividend down into the current dividend.
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Step 8.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 8.1.5.8
Multiply the new quotient term by the divisor.
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Step 8.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 8.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 8.1.5.11
Pull the next terms from the original dividend down into the current dividend.
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Step 8.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 8.1.5.13
Multiply the new quotient term by the divisor.
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Step 8.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 8.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 8.1.5.16
Since the remander is , the final answer is the quotient.
Step 8.1.6
Write as a set of factors.
Step 8.2
Factor using the AC method.
Step 8.2.1
Factor using the AC method.
Step 8.2.1.1
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 8.2.1.2
Write the factored form using these integers.
Step 8.2.2
Remove unnecessary parentheses.
Step 9
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10
Step 10.1
Set equal to .
Step 10.2
Add to both sides of the equation.
Step 11
Step 11.1
Set equal to .
Step 11.2
Add to both sides of the equation.
Step 12
Step 12.1
Set equal to .
Step 12.2
Subtract from both sides of the equation.
Step 13
The final solution is all the values that make true.
Step 14