# Algebra Examples

Identify the Zeros and Their Multiplicities
Step 1
Set equal to .
Step 2
Solve for .
Step 2.1
Factor the left side of the equation.
Step 2.1.1
Factor using the rational roots test.
Step 2.1.1.1
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 2.1.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 2.1.1.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
Step 2.1.1.3.1
Substitute into the polynomial.
Step 2.1.1.3.2
Raise to the power of .
Step 2.1.1.3.3
Raise to the power of .
Step 2.1.1.3.4
Multiply by .
Step 2.1.1.3.5
Subtract from .
Step 2.1.1.3.6
Multiply by .
Step 2.1.1.3.7
Subtract from .
Step 2.1.1.3.8
Step 2.1.1.4
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 2.1.1.5
Divide by .
Step 2.1.1.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
 - - - +
Step 2.1.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
 - - - +
Step 2.1.1.5.3
Multiply the new quotient term by the divisor.
 - - - + + -
Step 2.1.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
 - - - + - +
Step 2.1.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 - - - + - + -
Step 2.1.1.5.6
Pull the next terms from the original dividend down into the current dividend.
 - - - + - + - -
Step 2.1.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
 - - - - + - + - -
Step 2.1.1.5.8
Multiply the new quotient term by the divisor.
 - - - - + - + - - - +
Step 2.1.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
 - - - - + - + - - + -
Step 2.1.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 - - - - + - + - - + - -
Step 2.1.1.5.11
Pull the next terms from the original dividend down into the current dividend.
 - - - - + - + - - + - - +
Step 2.1.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
 - - - - - + - + - - + - - +
Step 2.1.1.5.13
Multiply the new quotient term by the divisor.
 - - - - - + - + - - + - - + - +
Step 2.1.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
 - - - - - + - + - - + - - + + -
Step 2.1.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 - - - - - + - + - - + - - + + -
Step 2.1.1.5.16
Since the remander is , the final answer is the quotient.
Step 2.1.1.6
Write as a set of factors.
Step 2.1.2
Factor using the AC method.
Step 2.1.2.1
Factor using the AC method.
Step 2.1.2.1.1
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 2.1.2.1.2
Write the factored form using these integers.
Step 2.1.2.2
Remove unnecessary parentheses.
Step 2.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3
Set equal to and solve for .
Step 2.3.1
Set equal to .
Step 2.3.2
Add to both sides of the equation.
Step 2.4
Set equal to and solve for .
Step 2.4.1
Set equal to .
Step 2.4.2
Add to both sides of the equation.
Step 2.5
Set equal to and solve for .
Step 2.5.1
Set equal to .
Step 2.5.2
Subtract from both sides of the equation.
Step 2.6
The final solution is all the values that make true. The multiplicity of a root is the number of times the root appears.
(Multiplicity of )
(Multiplicity of )
(Multiplicity of )
(Multiplicity of )
(Multiplicity of )
(Multiplicity of )
Step 3