Topic: Multiplying Polynomials

 Multiplying polynomials Distribute: Each term from one polynomial is multiplied by each term in the other polynomial. If there is more than one polynomial being multiplied, multiply two together first, then multiply the result by the remaining polynomial. Combine like terms.  Final answers should be written in standard form.

Every term in the first polynomial wants to “meet” every term already at the “polynomial party.” 3b meets everyone… Then –2c meets everyone… Then the like terms “hook up” to give us our final answer.

There are 2 terms in the first polynomial & 3 in the second. Set up a 2 x 3 square (yeah, I know it’s really a rectangle, but you get the point!). Set up your polynomials outside the table. Fill in the squares by multiplying the appropriate terms. Combine like terms & write your final answer.

We could rewrite this (x + y)(x + y)(x + y) & multiply it out… …or we could use Pascal’s Triangle!

 The triangle gives the coefficients for any binomial expansion (a + b) n, where n is a non-negative integer.  The first row represents n = 0, the second row n = 1, and so on.  Each row begins and ends with 1. Each term in between is the sum of the two terms above it.  The triangle can be expanded for any positive value of n.  PUT THIS TRIANGLE ON A NOTECARD

 For the expansion of any binomial (a + b) n, where n is a positive integer: The expansion will have n + 1 terms. The n th row determines the coefficient for each term in the expansion (remember the first row is n = 0). The exponent of a in the first term of the expansion is n. In each successive term, the exponent of a decreases by 1. The exponent of b in the first term of the expansion is 0. In each successive term, the exponent of b increases by 1. For each term in the expansion, a & b are raised to their respective powers & multiplied.  Let’s go back to our original problem…

Since n = 3, we will use the 4 th row of the table. The coefficient for the 1 st term is 1. the exponent for x is 3, the exponent for y is 0 (remember anything to the 0 power is 1). The coefficient for the 2 nd term is 3. The exponent for x decreases by 1, the exponent for y increases by 1 (notice the sum of the exponents will always = n). REMINDER: Exponents NEVER affect coefficients! The coefficient for the 3 rd term is 3. Continue the pattern for the exponents. The coefficient for the 4 th (& final) term is 1. The exponent for x is now 0 and the exponent for y is now 3. Therefore, the expansion of (x + y) 3 is What happens when we have numbers?

Since n = 4, we will use the 5 th row of the table. The coefficient for the 1 st term is 1. the exponent for y is 4, the exponent for -4 is 0. The coefficient for the 2 nd term is 4, the exponent for y is 3, the exponent for -4 is 1. The term must be simplified. We now have all 5 terms. The expansion of (y – 4) 4 is Continue this pattern for each successive term.

 TITLE: Checking My Understanding: Multiplying Polynomials  Review your notes from this presentation & create and complete the following subheadings in your journal: “Things I already knew:” Identify any information with which you were already familiar. “New things I learned:” Identify any new information that you now understand. “Questions I still have:” What do you still want to know or do not fully understand?

 Textbook Section 6-2 (pg. 419): 39-53