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# Multiplying Polynomials

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### Multiplying Polynomials

1. 1. ADDING, SUBTRACTING, & MULTIPLYING POLYNOMIALS
2. 2. ERROR ANALYSIS <ul><li>A student simplified 6a³ -a³ and got a 6 as a result. Write an explanation of the student’s error using the words coefficient and like terms. </li></ul>
3. 3. CHALLANGE <ul><li>A 35-centimeter stick is cut into three pieces so that the two end pieces are each equal in length to one-third of the middle piece. How long is each piece? </li></ul>
4. 4. MULTIPLYING POLYNOMIALS <ul><li>3a(6b + 7) </li></ul><ul><li>(3a)(6b) + (3a)(7) </li></ul><ul><li>18ab + 21a </li></ul><ul><li>2x(3x² + x – 4) </li></ul><ul><li>(2x)(3x²) + (2x)(x) + (2x)(-4) </li></ul><ul><li>6x³ + 2x² + -8x </li></ul>
5. 5. YOUR TURN <ul><li>-4a²(-2a² + 3ab – 2b + 5) </li></ul>
6. 6. MULTIPLYING BINOMIALS <ul><li>(x + 2)(x + 3) </li></ul><ul><li>x(x+ 3) + 2(x + 3) </li></ul><ul><li>(x)( x)+ (x)(3) + 2(x) + 2(3) </li></ul><ul><li>x² + 3x + 2x + 6 </li></ul><ul><li>x² + 5x + 6 </li></ul>
7. 7. FOIL METHOD <ul><li>(x + 3)(x + 2) + + + </li></ul><ul><li>x² + 2x + 3x + 6 </li></ul><ul><li>x² + 5x+ 6 </li></ul>first 2) outer (x inside last First x·x Outer x ·2 Inside 3 · x Last 3 · 2
8. 8. YOUR TURN – USE FOIL METHOD <ul><li>1. (x + 4)(x - 3) 2. (2x + 2)(x + 6) </li></ul><ul><li>3. (3ab² + b)(-2ab² - b) 4. (4xy + 3)(2x²y + 1) </li></ul>
9. 9. FOIL <ul><li>1. (x + 4)(x – 4) 2. (x + 3)(x – 3) </li></ul><ul><li>3. (3y – 1)(3y + 1) 4. (2x² + 3)(2x² - 3) </li></ul><ul><li>5. (2x +y) (2x -y) 6. (x²y - 2)(x²y + 2) </li></ul>
10. 10. FOIL <ul><li>1.(y + 1)(y + 1) 2. (x + 2)² </li></ul><ul><li>3. (2x - 3)² 4. .(y² + 1)² </li></ul><ul><li>5.(2x – y)² 6. .(2y² - 2x)² </li></ul>
11. 11. SOME BINOMIAL PRODUCTS APPEAR SO MUCH WE NEED TO RECOGNIZE THE PATTERNS! <ul><li>Sum & Difference (S&D): </li></ul><ul><li>(a + b)(a – b) = a 2 – b 2 </li></ul><ul><li>Example: (x + 3)(x – 3) = x 2 – 9 </li></ul><ul><li>Square of Binomial: </li></ul><ul><li>(a + b) 2 = a 2 + 2ab + b 2 </li></ul><ul><li>(a - b) 2 = a 2 – 2ab + b 2 </li></ul>
12. 12. MULTIPLYING POLYNOMIALS : HORIZONTALLY <ul><li>(3x 2 – 2x – 4)(x – 3) = </li></ul><ul><li>(3x 2 )(x – 3) </li></ul><ul><li>+ (-2x)(x – 3) </li></ul><ul><li>+ (-4)(x – 3) = </li></ul><ul><li>(3x 3 – 9x 2 ) + (-2x 2 + 6x) + (-4x + 12) = </li></ul><ul><li>3x 3 – 9x 2 – 2x 2 + 6x – 4x +12 = </li></ul><ul><li>3x 3 – 11x 2 + 2x + 12 </li></ul>
13. 13. MULTIPLYING POLYNOMIALS: VERTICALLY <ul><li>(-x 2 + 2x + 4)(x – 3)= </li></ul><ul><li>-x 2 + 2x + 4 * x – 3 </li></ul><ul><li> 3x 2 – 6x – 12 </li></ul><ul><li> -x 3 + 2x 2 + 4x_____ </li></ul><ul><li> -x 3 + 5x 2 – 2x – 12 </li></ul>
14. 14. MULTIPLYING 3 BINOMIALS : <ul><li>(x – 1)(x + 4)(x + 3) = </li></ul><ul><li>FOIL the first two: </li></ul><ul><li>(x 2 – x +4x – 4)(x + 3) = </li></ul><ul><li>(x 2 + 3x – 4)(x + 3) = </li></ul><ul><li>Then multiply the trinomial by the binomial </li></ul><ul><li>(x 2 + 3x – 4)(x) + (x 2 + 3x – 4)(3) = </li></ul><ul><li>(x 3 + 3x 2 – 4x) + (3x 2 + 9x – 12) = </li></ul><ul><li>x 3 + 6x 2 + 5x - 12 </li></ul>
15. 15. LAST PATTERN <ul><li>Cube of a Binomial </li></ul><ul><li>(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 </li></ul><ul><li>(a – b) 3 = a 3 - 3a 2 b + 3ab 2 – b 3 </li></ul>
16. 16. EXAMPLE: <ul><li>(x + 5) 3 = </li></ul><ul><li>a = x and b = 5 </li></ul><ul><li>x 3 + 3(x) 2 (5) + 3(x)(5) 2 + (5) 3 = </li></ul><ul><li>x 3 + 15x 2 + 75x + 125 </li></ul>