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Factoring Polynomials
1. Factoring Using the Distributive Prop
2. Difference of Two Squares
3. Difference of Two Cubes
4. Sum of Two Cubes
5. Trinomials
6. Grouping
2 term sD ifference of 2 sq uares
D ifferen ce of 2 cu b esS u m o f 2 cu b es
3 term sP erfec t S q uare?T rial and Error
4 term sG rou ping
D istributive PropertyC a n yo u d iv id e a ll te rm s b y the
sa m e nu m be r o r le tte r?
Factoring Polynomials
Distributive Property
5x2y3 – 15x3y + 25x2y
5x2y(y2 – 3x + 5)
2 Terms Difference of Squares
4x2 – 9y6
(2x – 3y3)(2x + 3y3)
2 Terms Difference of Cubes
8x3 – 27y6
(2x – 3y2)(4x2 + 6xy2 + 9y4) First Factor Second Factor - Use first factor
cube root of each term 1. Square 1st term
same sign 2. Change sign
3. Multiply the 2 terms
4. Square 2nd term (always +)
2 Terms Sum of Cubes
8x3 + 27y6
(2x + 3y2)(4x2 - 6xy2 + 9y4) First Factor Second Factor - Use first factor
cube root of each term 1. Square 1st term
same sign 2. Change sign
3. Multiply the 2 terms
4. Square 2nd term (always +)
3 Terms Perfect Square
4x2 – 20x + 25(2x - 5)2
* The first and last term must be perfect squaresa. Exponents have to be even to be perfect
squares1. Take the square root of the first term2. Take the first sign3. Take the square root of the last term.4. Check: the middle term should be 2 times the first term
times second term
Factoring Trinomials
When the leading coefficient is 1
Factoring Trinomials
1. Make sure trinomial is in descending order of variable.
Divide out any common factors.
Example:
x2 + 5x + 6
Factoring Trinomials
2. Start out with two sets of parentheses. These are the factors.
Example:
x2 + 5x + 6
( )( )
Factoring Trinomials
3. Put the variable given at the beginning of each factor
Example:
x2 + 5x + 6
( x )( x )
Factoring Trinomials
4A. Determine signs in factors.
a) Since the last sign is + the signs are the same
b) Since the first sign is + they are both +
x2 + 5x + 6
( x + )( x + )
Factoring Trinomials
4A. Find two factors of the last number that add up to the middle number.
Example:
x2 + 5x + 6
( x + 3 )( x + 2 )
Factoring Trinomials
4B. Determine signs in factors.
a) Since the last sign is + the signs are the same
b) Since the first sign is - they are both +
x2 - 5x + 6
( x - )( x - )
Factoring Trinomials
4B. Find two factors of the last number that add up to the middle number.
Example:
x2 - 5x + 6
( x - 3 )( x - 2 )
Factoring Trinomials
4C. Determine signs in factors.
a) Since the last sign is - the signs are different
b) Since the first sign is + the bigger number goes by the +
x2 + 5x - 6
( x + )( x - )
Factoring Trinomials
4C. Find two factors of the last number whose difference is the middle number.
Example:
x2 + 5x - 6
( x + 6 )( x - 1 )
Factoring Trinomials
4D. Determine signs in factors.
a) Since the last sign is - the signs are different
b) Since the first sign is - the bigger number goes by the +
x2 - 5x - 6
( x + )( x - )
Factoring Trinomials
4D. Find two factors of the last number whose difference is the middle number.
Example:
x2 - 5x - 6
( x + 1 )( x - 6 )
3 Terms Trial and Error (FOIL)
6x2 – 11x + 4(2x - 1)(3x - 4)
Signs:1. If last sign is + then both factors have the same sign
a. If the first sign is + both factors have + signb. If the first sign is – both factors have - sign
2. If last sign is – then both factors have different signs
4 Terms Grouping
x3 – 4x2 + 3x - 12x2(x - 4) + 3(x - 4)
(x – 4)(x2 + 3)