# Precalculus Examples

Find the Factors Using the Factor Theorem
,
Step 1
Divide using synthetic division and check if the remainder is equal to . If the remainder is equal to , it means that is a factor for . If the remainder is not equal to , it means that is not a factor for .
Step 1.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
Step 1.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
Step 1.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 1.4
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 1.5
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 1.6
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 1.7
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 1.8
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 1.9
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 1.10
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 1.11
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 1.12
Simplify the quotient polynomial.
Step 2
The remainder from dividing is , which means that is a factor for .
is a factor for
Step 3
Find all the possible roots for .
Step 3.1
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 3.2
Find every combination of . These are the possible roots of the polynomial function.
Step 4
Set up the next division to determine if is a factor of the polynomial .
Step 5
Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since divides evenly into , is a factor of the polynomial and there is a remaining polynomial of .
Step 5.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
Step 5.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
Step 5.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 5.4
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 5.5
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 5.6
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 5.7
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Step 5.8
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 5.9
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 5.10
Simplify the quotient polynomial.
Step 6
Find all the possible roots for .
Step 6.1
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 6.2
Find every combination of . These are the possible roots of the polynomial function.
Step 7
The final factor is the only factor left over from the synthetic division.
Step 8
The factored polynomial is .