Please open your laptops, log in to the MyMathLab course web site, and open Quiz 5.2. You may use the pink formula sheet on this quiz – please don’t write on this sheet, and remember to hand it back in with your quiz answer sheet.

Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note- taking materials.

Sections 5.3/5.4 Multiplying Polynomials

Multiplying polynomials If all of the polynomials are monomials, use the associative and commutative properties, along with properties of exponents. If any of the polynomials have more than one term, use the distributive property before the associative and commutative properties. Then combine like terms.

Example Multiply each of the following: 1) (3x 2 )(-2x) = (3 -2)(x 2 x) = -6x 3 2) (4x 2 )(3x 2 – 2x + 5) = (4x 2 )(3x 2 ) + (4x 2 )(-2x) + (4x 2 )(5) (distributive property) = 12x 4 – 8x x 2 (multiply the monomials) 3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5) (can also use “FOIL” on this) = 14x x – 28x – 20 = 14x 2 – 18x – 20

Example Multiply (3x + 4) 2 Remember that a 2 = a a, so (3x + 4) 2 = (3x + 4)(3x + 4). (3x + 4) 2 = (3x + 4)(3x + 4)= (3x)(3x + 4) + 4(3x + 4) = 9x x + 12x + 16 = 9x x + 16 EXTREMELY IMPORTANT NOTE: (3x + 4) 2 is NOT simply (3x) !!!

Example Multiply (a + 2)(a 3 – 3a 2 + 7). (a + 2)(a 3 – 3a 2 + 7) = a(a 3 – 3a 2 + 7) + 2(a 3 – 3a 2 + 7) = a 4 – 3a 3 + 7a + 2a 3 – 6a = a 4 – a 3 – 6a 2 + 7a + 14

Example Multiply (5x – 2z) 2 (5x – 2z) 2 = (5x – 2z)(5x – 2z)= (5x)(5x – 2z) – 2z(5x – 2z) = 25x 2 – 10xz – 10xz + 4z 2 = 25x 2 – 20xz + 4z 2 REMINDER: (5x -2z) 2 is NOT simply (5x) 2 – (2z) 2 !!!

Example Multiply (2x 2 + x – 1)(x 2 + 3x + 4) (2x 2 + x – 1)(x 2 + 3x + 4) = (2x 2 )(x 2 + 3x + 4) + x(x 2 + 3x + 4) – 1(x 2 + 3x + 4) = 2x 4 + 6x 3 + 8x 2 + x 3 + 3x 2 + 4x – x 2 – 3x – 4 = 2x 4 + 7x x 2 + x – 4

Special Products Some types of polynomial products can be carried out more efficiently using techniques that apply only to specific situations such as two binomials or squaring a binomial. These products can also be calculated using the basic laws of exponents and the distributive property, but these shortcuts may save you some time if you can learn to recognize the situations to which they apply. The shortcuts are based on the laws of exponents and the distributive property, as you will see as we go through the sample problems.

When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method. This is just a memory device that may be useful to keep you from forgetting any of the four parts of the product. FOIL only applies to a binomial (two-term polynomial) multiplied by another binomial, not to any other types of products involving monomials, trinomials, etc. F – product of First terms O – product of Outside terms I – product of Inside terms L – product of Last terms

= y 2 – 8y – 48 Example Multiply (y – 12)(y + 4) (y – 12)(y + 4) Product of First terms is y 2 Product of Outside terms is 4y Product of Inside terms is -12y Product of Last terms is -48 (y – 12)(y + 4) = F O I L y 2 + 4y – 12y – 48

Multiply (2x – 4)(7x + 5) Example (2x – 4)(7x + 5) = = 14x x – 28x – 20 F 2x(7x) F + 2x(5) O – 4(7x) I – 4(5) L O I L = 14x 2 – 18x – 20

In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products such as the following: Squaring a Binomial (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 (These might be just as easy to do by the usual FOIL method rather than by memorizing the formulas.) Multiplying the Sum and Difference of Two Terms (a + b)(a – b) = a 2 – b 2 (This formula can be quite useful and save you some time.) NOTE: These three formulas are on your formula sheet.

Problem from today’s homework:

Although you will arrive at the same results for the special products by using the distributive property, memorizing these products (especially the last one) can save you some time in multiplying polynomials. Multiplying 3 or more polynomials together might require you to use more than one technique. Multiply the polynomials two at a time.

Problem from today’s homework:

How would you attack this problem? (x – 5) 3 How about this problem? (2x – 3)(4x 2 + 9)(2x + 3)

REMINDER: The assignment on today’s material (HW 5.3/4) is due at the start of the next class session. Lab hours in 203: Mondays through Thursdays 8:00 a.m. to 7:30 p.m. Please remember to sign in on the Math 110 clipboard by the front door of the lab

You may now OPEN your LAPTOPS and begin working on the homework assignment. We expect all students to stay in the classroom to work on your homework till the end of the 55- minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice quiz/test.