Multiplying Binomials Section 8-3 Part 1 & 2

Goals Goal To multiply two binomials or a binomial by a trinomial. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary None

Multiplying Polynomials 3 Methods for multiplying polynomials 1.Using the Distributive Property Can be used to multiply any two polynomials 2.Using a Table or The Box Method Can be used to multiply any two polynomials 3.Using FOIL Can only be used to multiply two binomials

To multiply a binomial by a binomial, you can apply the Distributive Property more than once: (x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3. Distribute x and 3 again. Multiply. Combine like terms. = x(x + 2) + 3(x + 2) = x(x) + x(2) + 3(x) + 3(2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 Method 1: Distributive Property

Multiply. (s + 4)(s – 2) s(s – 2) + 4(s – 2) s(s) + s(–2) + 4(s) + 4(–2) s 2 – 2s + 4s – 8 s 2 + 2s – 8 Distribute s and 4. Distribute s and 4 again. Multiply. Combine like terms. Example: Multiply Using Distributive Property

Multiply. (a + 3)(a – 4) a(a – 4)+3(a – 4) a(a) + a(–4) + 3(a) + 3(–4) a 2 – a – 12 a 2 – 4a + 3a – 12 Distribute a and 3. Distribute a and 3 again. Multiply. Combine like terms. Your Turn:

+ 8 (y – 4) = (y 2 – 4y) = y 2 – 4y + 8y – 32 = y (y – 4) + (8y – 32) = y 2 + 4y – 32 Your Turn: Multiply. (y + 8)(y – 4)

Box method 2 x 2 parts = 2 rows 2 columns x + 6 2x +1 2x x + 1x+ 6 = 2x x + 1x + 6 = 2x x + 6 Visual model for distributing in polynomial products, works with any polynomial. Method 2: Box Method

Example: Multiply Using Box Method Multiply (x – 3)(4x – 5) 4x -5 x -3 4x x -12x = 4x 2 – 5x – 12x + 15 = 4x 2 – 17x + 15

Your Turn: Multiply (3x + 1)(x + 4) x +4 3x +1 3x x +x = 3x x + x + 4 = 3x x + 4

Your Turn: Multiply (2x - 5)(4x + 3) 4x +3 2x -5 8x x -20x = 8x 2 + 6x - 20x - 15 = 3x x - 15

The product can be simplified using the FOIL method: multiply the First terms, the Outer terms, the Inner terms, and the Last terms of the binomials. 2 FirstLast Inner Outer Method 3: FOIL

Holt Algebra Multiplying Polynomials 4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6  3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x  2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x  F O I L (x + 3)(x + 2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 F OIL 1. Multiply the First terms. (x + 3)(x + 2) x x = x 2  Example: Multiply Using FOIL “First Outer Inner Last”, shortcut for distributing, only works with binomial-binomial products. Multiply (x + 3)(x + 2)

= z (z)+ z (-12)-6 (z)-6 (-12) FOIL = z z– 6z+ 72 = z z + 72 = 5x (2x)+ 5x (8)-4 (2x)-4 (8) FOIL = 10x x– 8x – 32 = 10x x – 32 Example: FOIL

Multiply. A. (m – 2)(m – 8) B. (x + 3)(x + 4) (m – 2)(m – 8) (x + 3)(x + 4) m 2 – 8m – 2m + 16 x 2 + 4x + 3x + 12 FOIL m 2 – 10m +16x 2 + 7x +12 Your Turn:

Multiply. (x – 3)(x – 1) (x x) + (x(–1)) + (–3  x)+ (–3)(–1) ● x 2 – x – 3x + 3 x 2 – 4x + 3 Use the FOIL method. Multiply. Combine like terms. Your Turn:

Multiply. (2a – b 2 )(a + 4b 2 ) 2a(a) + 2a(4b 2 ) – b 2 (a) + (–b 2 )(4b 2 ) 2a 2 + 8ab 2 – ab 2 – 4b 4 2a 2 + 7ab 2 – 4b 4 Use the FOIL method. Multiply. Combine like terms. Your Turn:

To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x x – 6): (5x + 3)(2x x – 6) = 5x(2x x – 6) + 3(2x x – 6) = 5x(2x x – 6) + 3(2x x – 6) = 5x(2x 2 ) + 5x(10x) + 5x( – 6) + 3(2x 2 ) + 3(10x) + 3( – 6) = 10x x 2 – 30x + 6x x – 18 = 10x x 2 – 18

You can also use the Box Method to multiply polynomials with more than two terms. Multiply (5x + 3) by (2x x – 6): 2x22x2 +10x –6–6 10x 3 50x 2 – 30x 30x6x26x2 – 18 5x5x +3 Write the product of the monomials in each row and column: To find the product, add all of the terms inside the box by combining like terms and simplifying if necessary. 10x 3 + 6x x x – 30x – 18 10x x 2 – 18

Multiply. (x – 5)(x 2 + 4x – 6) x(x 2 + 4x – 6) – 5(x 2 + 4x – 6) x(x 2 ) + x(4x) + x(–6) – 5(x 2 ) – 5(4x) – 5(–6) x 3 + 4x 2 – 5x 2 – 6x – 20x + 30 x 3 – x 2 – 26x + 30 Distribute x and –5. Distribute x and −5 again. Simplify. Combine like terms. Example:

Multiply. (3x + 1)(x 3 + 4x 2 – 7) x3x3 4x24x2 –7–7 3x43x4 12x 3 – 21x 4x24x2 x3x3 –7–7 3x3x +1 3x x 3 + x 3 + 4x 2 – 21x – 7 Write the product of the monomials in each row and column. Add all terms inside the rectangle. 3x x 3 + 4x 2 – 21x – 7 Combine like terms. Your Turn:

Multiply. (x + 3)(x 2 – 4x + 6) x(x 2 – 4x + 6) + 3(x 2 – 4x + 6) Distribute x and 3. Distribute x and 3 again. x(x 2 ) + x(–4x) + x(6) +3(x 2 ) +3(–4x) +3(6) x 3 – 4x 2 + 3x 2 +6x – 12x + 18 x 3 – x 2 – 6x + 18 Simplify. Combine like terms. Your Turn:

= 3a(a 2 ) + 3a(-12a) + 3a(1) + 4(a 2 ) + 4(-12a) + 4(1) = 3a 3 – 36a 2 + 3a+ 4a 2 – 48a + 4 = 3a 3 – 32a 2 – 45a + 4 Box Method 3 x 3 terms = 3 by 3 box 2b 2 + 7b + 9 b 2 + 3b – 1 Combine like terms 2b 4 + 7b 3 + 9b 2 6b b b -2b 2 -7b -9= 2b b b b – 9 Your Turn:

The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. Write a polynomial that represents the area of the base of the prism. Write the formula for the area of a rectangle. Substitute h – 3 for w and h + 4 for l. A = l  w A = l w  A = (h + 4)(h – 3) Multiply. A = h 2 + 4h – 3h – 12 Combine like terms. A = h 2 + h – 12 The area is represented by h 2 + h – 12. Example: Application

The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. Find the area of the base when the height is 5 ft. A = h 2 + h – 12 A = – 12 A = – 12 A = 18 Write the formula for the area the base of the prism. Substitute 5 for h. Simplify. Combine terms. The area is 18 square feet. Your Turn:

Assignment 8-3 Part 1 Exercises Pg. 498: #4 – 22 even 8-3 Part 2 Exercises Pg. 502 – 503: #6 – 30 even