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Objective The student will be able to: multiply polynomials Unit 5: Quadratics

There are three techniques you can use for multiplying polynomials. The best part about it is that they are all the same! Huh? Whaddaya mean? It’s all about how you write it…Here they are! Distributive Property FOIL Box Method

Simplify: 4t2(3t2 + 2t - 5) Simplify: - 4m3(-3m - 6n + 4p) 12t4 + 8t3

Simplify: Simplify: 6rs(r2s - 3) 6a2 + 9a 6rs • r2s 6r3s2 - 18rs

Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8). Combine like terms. 10x2 + 31x + 24

The FOIL method is ONLY used when you multiply 2 binomials The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. All you’re doing is using the distributive property! Use the FOIL method to multiply the following binomials: (y + 3)(y + 7).

(y + 3)(y + 7). F tells you to multiply the FIRST terms of each binomial.

(y + 3)(y + 7). O tells you to multiply the OUTER terms of each binomial.

(y + 3)(y + 7). I tells you to multiply the INNER terms of each binomial. y2 + 7y + 3y

(y + 3)(y + 7). L tells you to multiply the LAST terms of each binomial. y2 + 7y + 3y + 21 Combine like terms. y2 + 10y + 21

Remember, FOIL reminds you to multiply the: First terms Outer terms Inner terms Last terms

The third method is the Box Method. This method works for every problem! Here’s how you do it. Multiply (3x – 5)(5x + 2) Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where. This will be modeled in the next problem along with FOIL. 3x -5 5x +2

Multiply (3x - 5)(5x + 2) 3x -5 5x +2 15x2 +6x -25x 15x2 -25x -10 +6x First terms: Outer terms: Inner terms: Last terms: Combine like terms. 15x2 - 19x – 10 3x -5 5x +2 +6x -25x 15x2 -25x -10 +6x -10 You have 3 techniques. Pick the one you like the best!

Multiply (7p - 2)(3p - 4) 7p -2 3p -4 21p2 -28p -6p 21p2 -6p +8 -28p First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p2 – 34p + 8 7p -2 3p -4 -28p -6p 21p2 -6p +8 -28p +8

Simplify 4y(3y2 – 1) 7y2 – 1 12y2 – 1 12y3 – 1 12y3 – 4y

Multiply (y + 4)(y – 3) y2 + y – 12 y2 – y – 12 y2 + 7y – 12

Simplify -3x2y3(y2 – x2 + 2xy) -3x2y5 + 3x4y3 – 6x3y4

Multiply (2a – 3b)(2a + 4b) 4a2 + 14ab – 12b2 4a2 – 14ab – 12b2 4a2 + 8ab – 6ba – 12b2 4a2 + 2ab – 12b2 4a2 – 2ab – 12b2

Group and combine like terms. Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x2 - 5x + 4) - 5(x2 - 5x + 4) 2x3 - 10x2 + 8x - 5x2 + 25x - 20 Group and combine like terms. 2x3 - 10x2 - 5x2 + 8x + 25x - 20 2x3 - 15x2 + 33x - 20

Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. x2 -5x +4 2x -5 2x3 -10x2 +8x Almost done! Go to the next slide! -5x2 +25x -20

Multiply (2x - 5)(x2 - 5x + 4) Combine like terms! +4 2x -5 2x3 -10x2 +8x -5x2 +25x -20 2x3 – 15x2 + 33x - 20

Multiply (2p + 1)(p2 – 3p + 4) 2p3 -5p2 + 5p + 4 p2 – p – 12 y2 – 7y – 12