3.5 (Part 1) Multiplying Two Binomials F 3.5 (Part 1) Multiplying Two Binomials O I Math 1201 L

To EXPAND two binomials we use a version of the distributive law called FOIL Step 1: Multiply the FIRST terms in the brackets. F

Step 2: Multiply the OUTSIDE terms.

Step 3: Multiply the INSIDE terms.

Step 4: Multiply the LAST terms.

Step 5: Collect like terms.

Example FOIL

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3.5 (Part 2) Factoring Trinomials in the form x2 + bx + c Math 1201

Factoring Simple Trinomials x2 + 10 x + 16 = (x + 2)(x + 8) Check by FOILing = x2 + 8x + 2x + 16 = x2 + 10x + 16 x2 + 9x + 20 x2 + 11x + 24 = (x + 5)(x + 4) = (x + 8)(x + 3) What relationship is there between product form and factored form? x2 + 5x + 4 = (x + 4)(x + 1)

Factoring Simple Trinomials Many trinomials can be written as the product of 2 binomials. Recall: (x + 4)(x + 3) = x2 + 3x + 4x + 12 = x2 + 7x + 12 The middle term of a simple trinomial is the SUM of the last two terms of the binomials. The last term of a simple trinomial is the PRODUCT of the last two terms of the binomials. Therefore this type of factoring is referred to as SUM-PRODUCT!

“What numbers multiply to the last term and add to the middle term?" To factor trinomials, you ask yourself… “What numbers multiply to the last term and add to the middle term?" x12 +7 x2 + 7x + 12 1,12 13 2,6 8 (x + 3)(x + 4) 3,4 7

x2 – 8x +12 Factor: ( x – 2)( x – 6) 1, 12 13 -1, -12 -13 2, 6 8 -2, -6 -8

m2 – 5m -14 (m + 2) (m – 7) Factor: -1, 14 13 1, -14 -13 -2, 7 5 2, -7 x (-14) -5 -1, 14 13 (m + 2) (m – 7) 1, -14 -13 -2, 7 5 2, -7 -5

Factor: x2 - 11x + 24 x2 + 13x + 36 x2 - 14x + 33

x2 + 12x + 32 x2 - 20x + 75 x2 + 4x – 45 x2 + 17x + 72 x2 - 7x – 8 Factor: x2 + 12x + 32 x2 - 20x + 75 x2 + 4x – 45 x2 + 17x + 72 x2 - 7x – 8

- 5t – 3t2 + 15 + 4t2 – 3 - 3t t2 – 8t +12 Factor: ( x – 2)( x – 6) STEP 1: Combine Like terms t2 – 8t +12 ( x – 2)( x – 6) x 12 - 8 1, 12 13 -1, -12 -13 2, 6 8 -2, -6 -8

7q2 – 14q - 21 7 ( q2 –2q –3) 7 ( q – 3)( q + 1) Factor: -3 -2 -1, 3 2 STEP 1: Pull out the GCF 7 ( q2 –2q –3) 7 ( q – 3)( q + 1) -3 -2 -1, 3 2 -3, 1 -2

To Summarize: Always check to see if you can simplify first! Then check to see if you can pull out a common factor. Write 2 sets of brackets with x in the first position. Find 2 numbers whose sum is the middle coefficient, and whose product is the last term. Check by foiling the factors. ex. + = 7 x = 10 common factor? 5, 2 ex. + = 1 x = -20 common factor? -4, 5

Lets take a look at x2 + 5x + 6 How could we factor this using algebra tiles? Create a rectangle using the exact number of tiles in the given expression. Remember that a trinomial represents area – two binomials multiplied together. What is the width and length of the rectangle? These are the FACTORS of the original rectangle. Does that make sense? (x+3)(x+2) x + 3 x + 2

Using algebra tiles, factor the following… Create a rectangle using the exact number of tiles in the given expression. Remember that a trinomial represents area – two binomials multiplied together. What is the width and length of the rectangle? These are the FACTORS of the original rectangle.

Homework Time!