Download Math 95 Review of Factoring

Math 95
Review of Factoring
Factor by grouping:
Factoring by grouping is useful when we have a polynomial with an even number of
terms. Four term polynomials are dealt with in this class.
1. Group terms that have a common factor. Sometimes the terms will need to be rearranged
2. Factor out the common factor from each group
3. Factor out the binomial factor (if one exists)
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Factor x + 5 x + 2 x + 10
Example:
2. Factor out the common factor from each group
x 3 + 2 x + 5 x 2 + 10
x x 2 + 2 + 5( x 2 + 2)
3. Factor out the binomial factor
(x + 5)(x 2 + 2)
1. Group the terms that have a common factor
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Factoring Trinomials whose Leading Coefficient is One
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Factor x + bx + c
1. Write down all the factors of c
2. Determine which factors add together to give b
3. Write the factors in the equation:
x+
(
)(x + )
4. Check your work by multiplying out the two binomials
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For example: Factor 2 x + 6 x − 56 x
First, factor the greatest common factor from each term
(
2 x 3 + 6 x 2 − 56 x = 2 x x 2 + 3 x − 28
)
The last factors, -4 and 7 add to equal 3, the coefficient of x, so:
2 x 3 + 6 x 2 − 56 x = 2 x( x − 4 )( x + 7 )
Factoring Trinomials by grouping.
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To factor ax + bx + c where a ≠ 1
1.
2.
3.
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Multiply the leading coefficient a, and the constant c
Find the factors of ac whose sum is b (the coefficient of x)
Rewrite the middle term, bx, as the sum or difference using the factors from step 2
Factor by grouping
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Example: factor 2 x − x − 6
ac = 2(− 6 ) = −12
The factors of –12 whose sum is –1 (the coefficient of x) are –4 and 3
Re-write the middle term:
2 x 2 − x − 6 = 2 x 2 − 4 x + 3x − 6
Factor by grouping
2 x 2 − 4 x + 3x − 6 =
2 x 2 − 4 x + (3 x − 6 ) =
2 x( x − 2 ) + 3( x − 2 ) =
(2 x + 3)( x − 2 )
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Factoring the difference of two squares
If A and B are real numbers, variables or algebraic expressions then
A 2 − B 2 = ( A − B )( A + B )
Example:
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)(
9 − 16 x10 = 3 + 4 x 5 3 − 4 x 5
Factoring the sum of two cubes
(
A3 + B 3 = ( A + B ) A 2 − AB + B 2
Factoring the difference of two cubes
(
A3 − B 3 = ( A − B ) A 2 + AB + B 2
)
)
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A General Factoring Strategy
1. If there is a common factor, factor out the GCF
2. Determine the number of terms in the polynomial and try to factor as follows:
a. If there are two terms, can the binomial be factored as:
i. The difference of two squares A − B = ( A − B )( A + B )
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ii. Sum of two cubes: A + B = ( A + B ) A − AB + B
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iii. Difference of two cubes A + B = ( A + B ) A − AB + B
b. If there are three terms, is the trinomial a perfect square trinomial?
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i. A + 2 AB + B = ( A + B )
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ii. A − 2 AB + B = ( A − B )
iii. If the trinomial is not a perfect square trinomial, try factoring by
trial and error, or by grouping
c. If there are four or more terms, try to factor by grouping
3. Check to see if any of the terms can be factored further
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