ab + ac = a(b + c) a 2 – b 2 = (a + b)(a – b)

Funny Pattern Rule. a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) 1. Outline: ( binomial ) ( trinomial ) ( ___ ___ ) ( ___ ___ ___ ) 2. S.O.A.P. the signs. SOAP SAME SIGN OPPOSITE SIGN ALWAYS PLUS 3. Cube root the terms for the binomial answer. 4. SMILE you’re almost done! This builds the trinomial. = ( ___ ___ ) ( ___ ___ ___ ) + – + x times x x times 3 3 times 3 See, you’re SMILING!

Sign Rules ( _ + _ – _ ) = ( _ + _ )( _ – _ ) ( _ – _ + _ ) = ( _ – _ )( _ – _ ) ( _ + _ + _ ) = ( _ + _ )( _ + _ ) ( _ – _ – _ ) = ( _ – _ )( _ + _ ) Trinomial ends in a minus, the binomials have opposite signs Trinomial ends in a plus, the binomials have same signs Lgsm Lg sm Rule to find factors for the binomials. Lgsm

We apply the same rules, but now the binomials are built by Factoring by Grouping. Cut the polynomial in half and remove the GCFs on the left side, the right side, and then both sides. 3-4 x15-8 Number Sense Rule for Adding: The only way to add to number together to get and odd number is an odd + an even! Find ODD factors in 10 and 12! 52 Number Sense Rule for Multiplying: (ODD)(ODD) = ODD Possible Factors… 3 and 40 = 5(2)(4) 5 and 24 = 2(3)(4) 15 and 8 = 2(4) Which pair add or subtract to 7? YES 15 – 8 NO GCF Left 15-8 x We now replace 7x with 15x – 8x and perform Factor by Grouping. 2x2x3 Binomials are always the same! 2x2x35x5x- 4 GCF Right GCF of both sides. (2x + 3)(5x – 4) ODD + EVEN = ODD NO x

Answer looks like. Refers to the middle term. ODD + EVEN = ODD odd even Since we have an odd + even, we need odd factors. Break the 10 and 12 down to odd factors. Isolate the odd factors and multiply all possible odd combinations. Not the factors Right factors It should still factor if we switch the 15x and -8x. I can see a pattern! When you look at the left side of each factoring by grouping, I see the two binomials! Do you see that? Say YES! What terms are generating these binomials? Look above each step. It is the leading term and the two factors! Can we all agree that we will always factor out at least an x as the GCF? Yep. Here is a shortcut. Always put the “a” in both binomials. Put in the factors. Take out GCF’s 25

Refers to the middle term. EVEN + EVEN = EVEN Answer looks like this using new short cut. Use 8x twice. even Because a = 8 Since we have an even + even, we factor out a 2 from our factors. Break the 8 down to get factors of 2’s. Put a 2( ) in each blank as a factor because we know that the two factors are even. 2( ) Factor 2 out of the -14. The sum of the two red ( )’s must = – 7. Since – 7 is odd. Isolate the odd factors and multiply all possible odd combinations. Right factors! Put them in the red ( )’s! 6 and -20 are the two factors that add up to -14. Place them in our answer. Now we know we are not finished because we used the 8 twice. We have to divide out the extra 8 by finding the GCF of each binomial. 24

EVEN RULE ( odd + odd ) Factor. Refers to the middle term. ODD + ODD = EVEN Answer looks like this using new short cut. Use 3x twice. odd odd Because both a & c are odd Since a and c as odd factors we have an odd + odd = -34. This is going to take some time because all the factors will be odd. Break 63 down. Isolate each odd factor, from smallest to largest, and then multiply all possible odd combinations to create more odd factors. Wrong factors!Right factors! -7 and -27 are the two factors that add up to -34. Place them in our answer. Now we know we are not finished because we used the 3 twice. We have to divide out the extra 3 by finding the GCF of each binomial. 3 NO GCF

GCF Left Sometimes we group our terms 3 to 1! Factor the side with 3 terms. x5 Binomials are always the same! x53x23x2 - 2 GCF Right GCF of both sides. (x + 5)(3x 2 – 2) Cut the polynomial in half and remove the GCFs on the left side, the right side, and then both sides. Creates Difference of Perfect Squares.

GCF of 5 Step 2 D.P.S. Step 2 D.P.S. Again GCF of 2x Step 4 F. by G. Step 2 D.P.S. GCF of 3 Step 4 F. by G. Step 2 D.P.S. & P.C. Step 2 D.P.S. Step 2 & P.C. twice

Factor completely. GCF of 7 Step 3 GCF of 3x 2 Step 3 3 to 1 SPLIT Step 4 F. by G. Difference of Per. Squares 1 to 3 SPLIT Step 4 F. by G. Difference of Per. Squares GCF of -1 first. Distribute the minus!

Remember the product of -8(10)(3)(5)(0)(7)(11) = 0. Solve each for x.

Solve the equations by factoring. We don’t have to list the same twice, but just know that there were two answers that were the same value. Never divide by the variable! Set the equation = 0. No reason to work out the 2 nd binomial because the only difference will be the sign.

Solve the equations by factoring. 2 odd even odd 3 The factors have to differ by 1, so 2(7)=14 and 3(5)=15

Solve the equations by factoring. even even even 2( ) One of the 3’s must be isolated, 3 and 18 will subtract to be 15 in the ( )’s. To save time we can solve these binomial right now. You will have to reduce.