1. Factoring Algebraic Expressions Finding Monomial Factors Ch. 10 10.1 & Multiplying Binomials Mentally Ch. 10 10.2

2. Copyright © Cengage Learning. All rights reserved. Finding Monomial Factors 10.1

3. Factoring an algebraic expression, like finding the prime factors of a number, means writing the expression as a product of factors. The prime factorization of 12 is 2  2  3. Other factorizations of 12 are 2  6 and 4  3. Since factorization means writing a number or an algebraic expression as a product, then a number or an algebraic expression divided by one factor generates another factor. Finding Monomial Factors

4. Thus, 12 divided by 2 gives 6, so 2 times 6 is a factorization of 12. To factor the expression 2x + 2y, notice that 2 is a factor common to both terms of the expression. In other words, 2 is a factor of 2x + 2y. To find the other factor, divide by 2. Therefore, a factorization of 2x + 2y is 2(x + y). Finding Monomial Factors

5. A monomial factor is a one-term factor that divides each term of an algebraic expression. Here, 2 divides each term of the algebraic expression and is called a monomial factor. When factoring any algebraic expression, always look first for monomial factors that are common to all terms. Finding Monomial Factors

6. Factor: 3a + 6b. First, look for a common monomial factor. Since 3 divides both 3a and 6b, 3 is a common monomial factor of 3a + 6b. Divide 3a + 6b by 3. = a + 2b Thus, 3a + 6b = 3(a + 2b). Check this result by multiplication: 3(a + 2b) = 3a + 6b. Example 1

7. Copyright © Cengage Learning. All rights reserved. Finding the Product of Two Binomials Mentally 10.2

8. We have learned how to multiply two binomials such as (2x + 3)(4x – 5) by the following method: 2x(4x) + 2x(-5) + 3(4x) + 3(-5) = (distribute) 8x 2 + (-10x) + 12x + (-15) =(combine like terms) 8x 2 + 2x – 15 Finding the Product of Two Binomials Mentally

9. This process of multiplying two binomials can be shortened as follows. Finding the Product of Two Binomials Mentally 1. The first term of the product is the product of the first terms of the binomials. 2. The middle term of the product is the sum of the outer product and the inner product of the binomials. 3. The last term of the product is the product of the last terms of the binomials. Finding the Product of Two Binomials Mentally

10. Let’s use the above steps to find the product (2x + 3)(4x – 5). Step 1: Product of the first terms: (2x)(4x) = 8x 2 Step 2: Outer product = (2x)(–5) = –10x (2x + 3)(4x – 5) Inner product = (3)(4x) = 12x Sum: 2x Step 3: Product of the last terms: (3)(–5) = – 15 Therefore, (2x + 3)(4x – 5) = 8x 2 + 2x – 15 Finding the Product of Two Binomials Mentally

11. Note that in each method, we found the exact same terms. The second method is much quicker, especially when you become more familiar and successful with it. The second method is used to factor polynomials. Therefore, it is very important that you learn to find the product of two binomials, mentally. This method is often called the FOIL method, where F refers to the product of the first terms, O refers to the outer product, I refers to the inner product, and L refers to the product of the last terms. Finding the Product of Two Binomials Mentally

12. Find the product (2x – 7)(3x – 4) mentally. Step 1: Product of the first terms (F): (2x)(3x) = 6x 2 Step 2: Outer (O) product = (2x)(–4) = –8x (2x – 7)(3x – 4) Inner (I) product = (–7)(3x) = –21x Sum: – 29x Step 3: Product of the last(F) terms: (–7)(–4) = 28 Therefore, (2x – 7)(3x – 4) = 6x 2 – 29x + 28 Example 1