4.4 Factoring Quadratic Expressions Learning Target: I can find common binomial factors of quadratic expressions. Success Criteria: I can find the factors of a quadratic

Multiply. (x+3)(x+2) x x + x x Multiplying Binomials (FOIL) FOIL = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 Distribute.

Let’s look at what each sign means as we factor. We will use x 2 - 7x + 12 as the first example. We want to look at the second sign first. The second sign tells us if the signs are the same or different. A “+” sign means they are the same and “–” means they are different. Now lets look at the first, if the signs are the same then the first sign tells us what both signs are when we factor. If they are different, the first sign tells us the sum factors. Factoring Trinomials

Again, we will factor trinomials such as x 2 + 7x + 12 back into binomials. We look for the pattern of products and sums! Factoring Trinomials If the x 2 term has no coefficient (other than 1)... Step 1: List all pairs of numbers that multiply to equal the constant, 12. x 2 + 7x = 1 12 = 2 6 = 3 4

Factoring Trinomials Step 2: Choose the pair that adds up to the middle coefficient. x 2 + 7x = 1 12 = 2 6 = 3 4 Step 3: Fill those numbers into the blanks in the binomials: ( x + )( x + ) 3 4 x 2 + 7x + 12 = ( x + 3)( x + 4)

Factor. x 2 + 2x - 24 This time, the constant is negative! Factoring Trinomials Step 1: List all pairs of numbers that multiply to equal the constant, -24. (To get -24, one number must be positive and one negative.) -24 = 1 -24, = 2 -12, = 3 -8, -3 8 = 4 -6, Step 2: Which pair adds up to 2? Step 3: Write the binomial factors. x 2 + 2x - 24 = ( x - 4)( x + 6)

Factor each trinomial, if possible. Watch out for signs!! 1) t 2 – 4t – 21 2) x x ) x 2 –10x ) - x 2 - 3x + 18 Factor These Trinomials!

Solution #1: t 2 – 4t – 21 1) Factors of -21: 1 -21, , ) Which pair adds to (- 4)? 3) Write the factors. t 2 – 4t – 21 = (t + 3)(t - 7)

Solution #2: x x ) Factors of 32: ) Which pair adds to 12 ? 3) Write the factors. x x + 32 = (x + 4)(x + 8)

Solution #3: x x )Factors of ) Which pair adds to -10 ? 3) Write the factors. x x + 24 = (x - 4)(x - 6) None of them adds to (-10). For the numbers to multiply to +24 and add to -10, they must both be negative!

Solution #4: -x 2 - 3x ) Factors of -18: 1 -18, , , ) Which pair adds to 3 ? 4) Write the factors. x 2 + 3x - 18 = -1 (x - 3)(x + 6) 1) Factors of -18:

The Factor Fish  Consider the following Trinomial: Y = 4x x – 12  This Trinomial has no common factors, it is not a perfect square and it is not a difference of squares … our only choice is “Guess and Check” … or is it?

The Factor Fish (Setup) Y = 4x x – 12 Begin by drawing the grid to the right on your paper.

The Factor Fish (Setup) Y = 4x x – 12 4x x Now follow the arrows to see where to put the values. Be sure to keep the signs for each term in the Trinomial.

The Factor Fish (Going Fishing) Y = 4x x – 12 4x x Next, Multiply the top two terms in the picture as shown. We are multiplying the leading term and the constant term.  = - 48x 2

The Factor Fish (Going Fishing) Y = 4x x – 12 4x x List all possible factors of -48. We are looking for factors that add to 13, so the larger numbers must be positive. - 48x 2

The Factor Fish (Going Fishing) Y = 4x x – 12 4x x Select the factors which add to the coefficient of our middle term (13). Input those values on the right side of the fish. - 48x 2 -3x16x + 

The Factor Fish (Going Fishing) Y = 4x x – 12 4x x The rest of the work is finding common factors of monomials 4x 2 -3x What is the Greatest Common Factor of both 4x 2 and -3x? - 48x 2 -3x16x ?

The Factor Fish (Going Fishing) Y = 4x x – 12 4x x Now find the rest of the common factors in each row and column - 48x 2 -3x16x x ? ??

The Factor Fish (Going Fishing) Y = 4x x – 12 4x x Notice how each of the inner rows and columns work now. = 4x 2 x × 4x = 4x 2 × = × 4 = -12 × = -3x x × -3 = -3x × = 16x 4x × 4 = 16x - 48x 2 -3x16x x-3 4x4

The Factor Fish (The Answer) Initial Trinomial Y = 4x x – 12 Factored Y = (x + 4) × (4x – 3) 4x x - 48x 2 -3x16x x-3 4x4

Remember  Look for things to factor out right away  3y y + 45  Factors to ….  3 (y 2 + 8y + 15)  Then  3(y+5)(y+3)

Homework  P. 221 #15-31 odds and odds   Challenge = #84

Factor. 3x x + 8 This time, the x 2 term DOES have a coefficient (other than 1)! Factoring Trinomials ( Method 2* ) Step 2: List all pairs of numbers that multiply to equal that product, = 1 24 = 2 12 = 3 8 = 4 6 Step 3: Which pair adds up to 14? Step 1: Multiply 3 8 = 24 (the leading coefficient & constant).

( 3x + 2 )( x + 4 ) 2 Factor. 3x x + 8 Factoring Trinomials ( Method 2* ) Step 5: Put the original leading coefficient (3) under both numbers. ( x + )( x + ) Step 6: Reduce the fractions, if possible. Step 7: Move denominators in front of x. Step 4: Write temporary factors with the two numbers ( x + )( x + ) ( x + )( x + ) 4 3

( 3x + 2 )( x + 4 ) Factor. 3x x + 8 Factoring Trinomials ( Method 2* ) You should always check the factors by distributing, especially since this process has more than a couple of steps. = 3x x + 8 = 3x x + 3x x √ 3x x + 8 = (3x + 2)(x + 4)

Factor 3x x + 4 This time, the x 2 term DOES have a coefficient (other than 1)! Factoring Trinomials ( Method 2* ) Step 2: List all pairs of numbers that multiply to equal that product, = 1 12 = 2 6 = 3 4 Step 3: Which pair adds up to 11? Step 1: Multiply 3 4 = 12 (the leading coefficient & constant). None of the pairs add up to 11, this trinomial can’t be factored; it is PRIME.

Try this one #5: 2x 2 + x ) Multiply 2 (-21) = - 42; list factors of , , , , ) Which pair adds to 1 ? 3) Write the temporary factors. 2x 2 + x - 21 = (x - 3)(2x + 7) ( x - 6)( x + 7) 4) Put “2” underneath ) Reduce (if possible). ( x - 6)( x + 7) ) Move denominator(s)in front of “x”. ( x - 3)( 2x + 7)

Try this one too #6: 3x x ) Multiply 3 10 = 30; list factors of ) Which pair adds to 11 ? 3) Write the temporary factors. 3x x + 10 = (3x + 5)(x + 2) ( x + 5)( x + 6) 4) Put “3” underneath ) Reduce (if possible). ( x + 5)( x + 6) ) Move denominator(s)in front of “x”. ( 3x + 5)( x + 2)