Concept 2 Difference of Squares

Make one addition and one subtraction. The degree is two (a quadratic function) The number of terms must be 2 (a binomial) Must be able to take the square root of a and C. *****Must be a minus sign! *** Take the square root of the first term. This goes first in both parentheses. Take the square root of the last term. This goes in the back of both parentheses. Make one addition and one subtraction.

1. 𝑥 2 −16 𝑥 2 = x + − x 4 x 4 16 = 4

2. 81𝑥 2 −36 + − 9x 9x 6 6

3. 16𝑥 2 −49 + − 4x 4x 7 7

4. 8𝑥 2 −25 + − 8 has no square root, so it is not factorable

5. 36𝑥 2 −225 6𝑥 + 15 6𝑥 −15

6. 8𝑥 2 −50 2 4𝑥 2 −25 2 2𝑥 +5 2𝑥 −5

7. 9𝑥 3 −81𝑥 9𝑥 𝑥 2 −9 9𝑥 𝑥 +3 𝑥 −3

8. 𝑥 2 −18𝑥+81 − − 𝑥− 9 𝑥− 9 𝑥− 9 2 Factoring Sum of Squares When the middle term is double the square root of c. − − 81 =9 𝑥− 9 𝑥− 9 9 ∗2=18 𝑥− 9 2 This will mean the signs are the same and whatever is in front of the b term.

9. 9𝑥 2 −30𝑥+25 3𝑥− 5 3𝑥−5 3𝑥− 5 2 Factoring Sum of Squares When the middle term is double the square root of c. 3𝑥− 5 3𝑥−5 9 𝑥 2 =3𝑥 25 =5 3𝑥− 5 2 3𝑥∗5 ∗2=30 This will mean the signs are the same and whatever is in front of the b term.

Factoring Sum of Squares 10. 8𝑥 2 +48𝑥+72 2 4 𝑥 2 +24𝑥+36 2 2𝑥+ 6 2𝑥+6 2 2𝑥+ 6 2

Factoring Polynomials When ….. Concept 3

1 The degree is two (a quadratic function) The number of terms must be 3 (a trinomial) The coefficient of the squared term is 1. 1

Draw parentheses and fill in your variables. Make a list of factors of c. Chose the set that add to b. Give your answer in the parenthesis. Check your work by distributing (foil). Make sure all your signs are correct.

Examples 1. 𝑥 2 +4𝑥+3 3. 𝑥 2 −3𝑥+2 (𝑥− )(𝑥− ) (𝑥+ )(𝑥+ ) (𝑥−1 )(𝑥−2 ) The C is positive so both signs are the same and they will be the sign in front of b which is +. 1. 𝑥 2 +4𝑥+3 3. 𝑥 2 −3𝑥+2 (𝑥− )(𝑥− ) (𝑥+ )(𝑥+ ) Factors of 3 are: 1 & 3 (𝑥−1 )(𝑥−2 ) (𝑥+1 )(𝑥+3 ) Signs are the same so factors add to get the b. 1 + 3 = 4 2. 𝑥 2 −5𝑥+6 The C is positive so both signs are the same and they will be the sign in front of b which is – . 4. 𝑥 2 +8𝑥+12 (𝑥− )(𝑥− ) (𝑥+ )(𝑥+ ) Factors of 6 are: 1 & 6 2 & 3 (𝑥−2 )(𝑥−3) (𝑥+2 )(𝑥+6 ) Signs are the same so factors add to get the b. 2 + 3 = 5

More Examples 5. 𝑥 2 +3𝑥−18 7. 𝑥 2 −2𝑥−24 (𝑥+ )(𝑥− ) (𝑥+ )(𝑥− ) The C is negative so the signs are different. 5. 𝑥 2 +3𝑥−18 7. 𝑥 2 −2𝑥−24 Factors of 18 are: 1 & 18 3 & 6 2 & 9 (𝑥+ )(𝑥− ) (𝑥+ )(𝑥− ) (𝑥+6 )(𝑥−3 ) Signs are different so factors subtract to get the b. Pay attention to the sign of b. 6 – 3 = 3 The one with the minus sign stays with it in the parentheses. 8. 2𝑥 2 −10𝑥−24 6. 6𝑥 2 +42𝑥−108 Remember to factor out any common factors first. 2( 𝑥 2 +5𝑥−12) 6( 𝑥 2 +7𝑥−18) Signs are different so factors subtract to get the b. Pay attention to the sign of b. 9 – 2 = 7 2(𝑥+ )(𝑥− ) 6(𝑥+ )(𝑥− ) 6(𝑥+9 )(𝑥−2 )

The degree is two (a quadratic function) The number of terms must be 3 (a trinomial) The coefficient of the squared term is >1 .

SLIDE – Multiply a and c and rewrite expression SLIDE – Multiply a and c and rewrite expression. (make sure first there is not common factors for a, b, and c.) FACTOR DIVIDE – Divide by a and reduce any fractions. BOTTOMS UP – bring up an remaining bottoms of a f fraction.

Examples 1. 7𝑥 2 +29𝑥+4 2. 4𝑥 2 +22𝑥+10 2(2𝑥 2 +11𝑥+5) 𝑥 2 +29𝑥+28 1. 7𝑥 2 +29𝑥+4 2. 4𝑥 2 +22𝑥+10 Slide the a. 7 * 4 = 28 2(2𝑥 2 +11𝑥+5) 𝑥 2 +29𝑥+28 Signs are the same and both positive. Add to get the middle number 2(𝑥 2 +11𝑥+10) (𝑥 + )(𝑥 + ) 1 & 28 2 & 14 4 & 7 2(𝑥 + 1)(𝑥 +10) (𝑥 +1)(𝑥 +28) 2 2 7 7 Now divide by a, which was 7 2(2𝑥 +1)(𝑥 +5) Now simplify if possible. Any fractions get moved up to be the number in front of x. (7𝑥 +1)(𝑥 +4)

More Examples 3. 2𝑥 2 −3𝑥−9 4. 3𝑥 2 +2𝑥−16 𝑥 2 +2𝑥−48 𝑥 2 −3𝑥−18 1 & 48 2 & 24 3 & 16 4 & 12 6 & 8 3. 2𝑥 2 −3𝑥−9 4. 3𝑥 2 +2𝑥−16 Slide the a. 2* 9 = 18 𝑥 2 +2𝑥−48 𝑥 2 −3𝑥−18 Signs are different. Subtract to get the middle number (𝑥 + )(𝑥 − ) (𝑥 + )(𝑥 − ) 1 & 18 2 & 9 3 & 6 (𝑥 +8)(𝑥 −6) (𝑥 +3)(𝑥 −6) 3 3 3 – 6 = -3 2 2 Now divide by a, which was 2 (3𝑥 +8)(𝑥 −2) (2𝑥 +3)(𝑥 −3) Now simplify if possible. Any fractions get moved up to be the number in front of x.

More Examples 5. 4𝑥 2 +38𝑥+70 2(2𝑥 2 +19𝑥+35) 2(𝑥 2 +19𝑥+70) 1 & 70 2 & 35 5 & 14 7 & 10 5. 4𝑥 2 +38𝑥+70 2(2𝑥 2 +19𝑥+35) 2(𝑥 2 +19𝑥+70) 2(𝑥 + )(𝑥+ ) 2(𝑥 +14)(𝑥+5) 2 2 2(𝑥 +7)(2𝑥+5)