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)