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# Notes 12.1 multiplying polynomials

## by Lori Rapp, Teacher at NCVPS on Apr 11, 2011

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## Notes 12.1 multiplying polynomialsPresentation Transcript

• MultiplyingPolynomials
• Table of ContentsSlide 3-4: Multiplying a Polynomial by a MonomialSlide 5-14: Practice Multiplying a Polynomial by aMonomialSlide 15: Multiplying Polynomials using thehorizontal and vertical methodsSlide 16: Multiplying Polynomials using the BoxMethodSlide 17-22: Practice Multiplying Polynomials
• Multiply a Polynomial by a MonomialMultiply each terminside the parenthesis 2 ( 2 3x 2x − 7x + 5 )by the monomialoutside theparenthesis.The number of termsinside the parenthesiswill be the same asafter multiplying.
• Multiply a Polynomial by a MonomialMultiply each terminside the parenthesis 2 ( 3x 2x − 7x + 52 )by the monomialoutside the 3x 2 ( 2x ) + 3x ( −7x ) + 3x ( 5 ) 2 2 2parenthesis.The number of termsinside the parenthesiswill be the same asafter multiplying.
• Multiply a Polynomial by a MonomialMultiply each terminside the parenthesis 2 ( 3x 2x − 7x + 52 )by the monomialoutside the 3x 2 ( 2x ) + 3x ( −7x ) + 3x ( 5 ) 2 2 2parenthesis.The number of termsinside the parenthesiswill be the same asafter multiplying.
• Multiply a Polynomial by a MonomialMultiply each terminside the parenthesis 2 ( 3x 2x − 7x + 52 )by the monomialoutside the 3x 2 ( 2x ) + 3x ( −7x ) + 3x ( 5 ) 2 2 2parenthesis.The number of termsinside the parenthesis 2 ( 3x 2x − 7x + 5 2 )will be the same asafter multiplying.
• Multiply a Polynomial by a MonomialMultiply each terminside the parenthesis 2 ( 3x 2x − 7x + 52 )by the monomialoutside the 3x 2 ( 2x ) + 3x ( −7x ) + 3x ( 5 ) 2 2 2parenthesis.The number of termsinside the parenthesis 2 ( 3x 2x − 7x + 5 2 )will be the same as 4 6x − 21x + 15x 3 2after multiplying.
• Multiply a Polynomial by a MonomialReview this Cool Math site to learn aboutmultiplying a polynomial by a monomial.Do the Try It and Your Turn problems inyour notebook and check your answers onthe next slides.
• Cool Math Try It - Page 1 Multiply: 4 ( 6x 2x + 3 2 )
• Cool Math Try It - Page 1 Multiply: 4 ( 6x 2x + 32 )Distribute the monomial.
• Cool Math Try It - Page 1 Multiply: 4 ( 6x 2x + 3 2 )Distribute the monomial. 4 2 4 6x ⋅ 2x + 6x ⋅ 3
• Cool Math Try It - Page 1 Multiply: 4 ( 6x 2x + 3 2 )Distribute the monomial. 4 2 4 6x ⋅ 2x + 6x ⋅ 3 Multiply each term.
• Cool Math Try It - Page 1 Multiply: 4 ( 6x 2x + 3 2 )Distribute the monomial. 4 2 4 6x ⋅ 2x + 6x ⋅ 3 Multiply each term. 6 4 12x + 18x
• Cool Math Try It - Page 1 Multiply: 4 ( 6x 2x + 3 2 )Distribute the monomial. 4 2 4 6x ⋅ 2x + 6x ⋅ 3 Multiply each term. 6 4 12x + 18x Verify that your answer has same number of terms as inside original ( ). Both have 2 terms.
• What is the degree of the previous answer? 6 4 12x + 18x
• What is the degree of the previous answer? 6 4 12x + 18xFirst term is degree 6.
• What is the degree of the previous answer? 6 4 12x + 18xFirst term is degree 6.Second term is degree 4.
• What is the degree of the previous answer? 6 4 12x + 18xFirst term is degree 6.Second term is degree 4.Therefore, the polynomial is degree 6.
• Your Turn - Page 2 multiply:
• Your Turn - Page 2 multiply: 3 ( 5 2 10x 2x + 1 − 3x + x )
• Your Turn - Page 2 multiply: 3 ( 5 2 10x 2x + 1 − 3x + x )Distribute the monomial.
• Your Turn - Page 2 multiply: 3 ( 10x 2x + 1 − 3x + x5 2 )Distribute the monomial. ( ) 10x 2x + 10x (1) + 10x −3x + 10x ( x ) 3 5 3 3 ( 2 ) 3
• Your Turn - Page 2 multiply: 3 ( 10x 2x + 1 − 3x + x5 2 ) ( ) 10x 2x + 10x (1) + 10x −3x + 10x ( x ) 3 5 3 3 ( 2 ) 3Multiply each term.
• Your Turn - Page 2 multiply: 3 ( 10x 2x + 1 − 3x + x 5 2 ) ( ) 10x 2x + 10x (1) + 10x −3x + 10x ( x ) 3 5 3 3 ( 2 ) 3Multiply each term. 8 3 5 4 20x + 10x − 30x + 10x
• Your Turn - Page 2 multiply: 3 ( 10x 2x + 1 − 3x + x 5 2 ) ( ) 10x 2x + 10x (1) + 10x −3x + 10x ( x ) 3 5 3 3 ( 2 ) 3 8 3 5 4 Put in descending 20x + 10x − 30x + 10x order and verify number of terms.(Both have 4 terms.)
• Your Turn - Page 2 multiply: 3 ( 10x 2x + 1 − 3x + x 5 2 ) ( ) 10x 2x + 10x (1) + 10x −3x + 10x ( x ) 3 5 3 3 ( 2 ) 3 8 3 5 4 Put in descending 20x + 10x − 30x + 10x order and verify number of terms. 8 5 4 3(Both have 4 terms.) 20x − 30x + 10x + 10x
• What is the degree of the previous answer? 8 5 4 3 20x − 30x + 10x + 10x
• What is the degree of the previous answer? 8 5 4 3 20x − 30x + 10x + 10xFirst term is degree 8.
• What is the degree of the previous answer? 8 5 4 3 20x − 30x + 10x + 10xFirst term is degree 8.Second term is degree 5.
• What is the degree of the previous answer? 8 5 4 3 20x − 30x + 10x + 10xFirst term is degree 8.Second term is degree 5.Third term is degree 4.
• What is the degree of the previous answer? 8 5 4 3 20x − 30x + 10x + 10xFirst term is degree 8.Second term is degree 5.Third term is degree 4.Fourth term is degree 3.
• What is the degree of the previous answer? 8 5 4 3 20x − 30x + 10x + 10xFirst term is degree 8.Second term is degree 5.Third term is degree 4.Fourth term is degree 3.Therefore, the polynomial is degree 8.
• Try It - Page 2 Multiply: 2 5 ( 2 2 4 4x w w − x + 6xw − 1 + 3x w 8 )
• Try It - Page 2 Multiply: 2 5 ( 2 2 4 4x w w − x + 6xw − 1 + 3x w 8 )Distribute the monomial.
• Try It - Page 2 Multiply: 2 5 ( 4x w w − x + 6xw − 1 + 3x w 2 2 4 8 )Distribute the monomial. 5 ( 2 ) 2 5 ( 2 ) 4x w ( w ) + 4x w −x + 4x w 6xw + 4x w ( −1) + 4x w 3x w 2 5 2 2 5 2 5 ( 4 8 )
• Try It - Page 2 Multiply: 2 5 ( 4x w w − x + 6xw − 1 + 3x w 2 2 4 8 ) 5 ( 2 ) 2 5 ( 2 )4x w ( w ) + 4x w −x + 4x w 6xw + 4x w ( −1) + 4x w 3x w 2 5 2 2 5 2 5 ( 4 8 )Multiply each term.
• Try It - Page 2 Multiply: 2 5 ( 4x w w − x + 6xw − 1 + 3x w 2 2 4 8 ) 5 ( 2 )4x w ( w ) + 4x w −x + 4x w 6xw + 4x w ( −1) + 4x w 3x w 2 5 2 2 5 ( 2 ) 2 5 2 5 ( 4 8 )Multiply each term. 2 6 4 5 3 7 2 5 6 13 4x w − 4x w + 24x w − 4x w + 12x wVerify answer has 5 terms like original parenthesis.
• What is the degree of the previous answer? 2 6 4 5 3 7 2 5 6 134x w − 4x w + 24x w − 4x w + 12x w
• What is the degree of the previous answer? 2 6 4 5 3 7 2 5 6 134x w − 4x w + 24x w − 4x w + 12x wFirst term is degree 8.
• What is the degree of the previous answer? 2 6 4 5 3 7 2 5 6 134x w − 4x w + 24x w − 4x w + 12x wFirst term is degree 8.Second term is degree 9.
• What is the degree of the previous answer? 2 6 4 5 3 7 2 5 6 134x w − 4x w + 24x w − 4x w + 12x wFirst term is degree 8.Second term is degree 9.Third term is degree 10.
• What is the degree of the previous answer? 2 6 4 5 3 7 2 5 6 134x w − 4x w + 24x w − 4x w + 12x wFirst term is degree 8.Second term is degree 9.Third term is degree 10.Fourth term is degree 7.
• What is the degree of the previous answer? 2 6 4 5 3 7 2 5 6 134x w − 4x w + 24x w − 4x w + 12x wFirst term is degree 8.Second term is degree 9.Third term is degree 10.Fourth term is degree 7.Fifth term is degree 19.
• What is the degree of the previous answer? 2 6 4 5 3 7 2 5 6 134x w − 4x w + 24x w − 4x w + 12x wFirst term is degree 8.Second term is degree 9.Third term is degree 10.Fourth term is degree 7.Fifth term is degree 19.Therefore, the polynomial is degree 19.
• Try this one... Multiply: ( 2 3x 2x − 5x + 7 )
• Try this one... Multiply: ( 2 3x 2x − 5x + 7 )Distribute the monomial.
• Try this one... Multiply: ( 2 3x 2x − 5x + 7 )Distribute the monomial. ( ) 3x 2x + 3x ⋅ ( −5x ) + 3x ( 7 ) 2
• Try this one... Multiply: ( 2 3x 2x − 5x + 7 ) ( ) 3x 2x + 3x ⋅ ( −5x ) + 3x ( 7 ) 2Multiply each term.
• Try this one... Multiply: ( 2 3x 2x − 5x + 7 ) ( ) 3x 2x + 3x ⋅ ( −5x ) + 3x ( 7 ) 2Multiply each term. 3 2 6x − 15x + 21x
• What is the degree of the previous answer? 3 2 6x − 15x + 21x
• What is the degree of the previous answer? 3 2 6x − 15x + 21xFirst term is degree 3.
• What is the degree of the previous answer? 3 2 6x − 15x + 21xFirst term is degree 3.Second term is degree 2.
• What is the degree of the previous answer? 3 2 6x − 15x + 21xFirst term is degree 3.Second term is degree 2.Third term is degree 1.
• What is the degree of the previous answer? 3 2 6x − 15x + 21xFirst term is degree 3.Second term is degree 2.Third term is degree 1.Therefore, the polynomial is degree 3.
• Try this one... Multiply: 2 2 ( 3 −2a b a + 3a b − 4b2 3 5 )
• Try this one... Multiply: 2 2 ( 3 −2a b a + 3a b − 4b 2 3 5 )Distribute the monomial.
• Try this one... Multiply: 2 2 ( 3 −2a b a + 3a b − 4b 2 3 5 )Distribute the monomial. ( −2a b )( a ) + ( −2a b )( 3a b ) + ( −2a b )( −4b ) 2 2 3 2 2 2 3 2 2 5
• Try this one... Multiply: 2 2 ( 3 −2a b a + 3a b − 4b 2 3 5 )( −2a b )( a ) + ( −2a b )( 3a b ) + ( −2a b )( −4b ) 2 2 3 2 2 2 3 2 2 5Multiply each term.
• Try this one... Multiply: 2 2 ( 3 −2a b a + 3a b − 4b 2 3 5 )( −2a b )( a ) + ( −2a b )( 3a b ) + ( −2a b )( −4b ) 2 2 3 2 2 2 3 2 2 5Multiply each term. 5 2 4 5 2 7 −2a b − 6a b + 8a b
• What is the degree of the previous answer? 5 2 4 5 2 7 −2a b − 6a b + 8a b
• What is the degree of the previous answer? 5 2 4 5 2 7 −2a b − 6a b + 8a bFirst term is degree 7.
• What is the degree of the previous answer? 5 2 4 5 2 7 −2a b − 6a b + 8a bFirst term is degree 7.Second term is degree 9.
• What is the degree of the previous answer? 5 2 4 5 2 7 −2a b − 6a b + 8a bFirst term is degree 7.Second term is degree 9.Third term is degree 9.
• What is the degree of the previous answer? 5 2 4 5 2 7 −2a b − 6a b + 8a bFirst term is degree 7.Second term is degree 9.Third term is degree 9.Therefore, the polynomial is degree 9.
• Multiplying PolynomialsWatch this 6 minute video to learn how to multiply atrinomial by a binomial.Here’s the link to copy/paste if the hyperlink didn’t work: http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-270s.htmlThe video shows you 2 methods, the horizontal methodand vertical method.Alternative: Visit the PurpleMath website to learn howto multiply polynomials using these methods.The next slide will show you another method formultiplying polynomials, called the box method.
• Box methodThe previous video showed you how tomultiply 2 polynomials, which can get messy.The Box Method is a way to keep youorganized while multiplying.Follow this link to see a 5 minute videoorganizing the multiplication using boxes.Here’s the link to copy/paste if the hyperlink doesn’t work: http://www.slideshare.net/secret/iiYvYrvk1SxdrG
• Practice Multiplying 2 Binomials You’ve seen 3 different methods for multiplying polynomial: 1) Horizontal Method; 2) Vertical Method; 3) Box Method Practice your favorite method at Coolmath. Select the “Give me a Problem” button to keep trying problems. Do your work in a notebook. When you select “What’s the Answer?” your answer is erased and correct answer is displayed. Having your work in a notebook will allow you to compare your answer to the correct answer. Keep working problems until you get 4 out of 5 correct. The next 2 slides show multiplying 2 binomials using the box method.
• Example: ( 5x + 8 ) ( 3x − 1)
• Example: ( 5x + 8 ) ( 3x − 1) 3x −15x8
• Example: ( 5x + 8 ) ( 3x − 1) 3x −15x 15x 2 5x8 24x −8
• Example: ( 5x + 8 ) ( 3x − 1) 2 3x −1 = 15x + 29x − 85x 15x 2 5x8 24x −8
• Example: ( 4n − 3) ( 3n − 2 )
• Example: ( 4n − 3) ( 3n − 2 ) 3n −24n−3
• Example: ( 4n − 3) ( 3n − 2 ) 3n −24n 12n 2 −8n−3 −9n 6
• Example: ( 4n − 3) ( 3n − 2 ) 2 3n −2 = 12n − 17n + 64n 12n 2 −8n−3 −9n 6
• Practice Multiplying 2 Polynomials Now that you are an EXPERT at the easy problems, try some harder problems at Coolmath. If you have trouble, go back and review a method. Remember, you can also see me on Pronto! Select the “Give me a Problem” button to keep trying problems. Do your work in a notebook. When you select “What’s the Answer?” your answer is erased and correct answer is displayed. Having your work in a notebook will allow you to compare your answer to the correct answer. Keep working problems until you get 4 out of 5 correct. The next 2 slides show multiplying 2 polynomials using the box method.
• Example: ( 4k + 3k + 9 ) ( k + 3) 2
• Example: ( 4k + 3k + 9 ) ( k + 3) 2 2 4k 3k 9k3
• Example: ( 4k + 3k + 9 ) ( k + 3) 2 2 4k 3k 9k 3 2 4k 3k 9k3 12k 2 9k 27
• Example: ( 4k + 3k + 9 ) ( k + 3) 2 2 4k 3k 9k 3 2 4k 3k 9k3 12k 2 9k 27 3 2 = 4k + 15k + 18k + 27
• Example: ( 2x − 3) ( 3x − 5x + 7 ) 2
• Example: ( 2x − 3) ( 3x − 5x + 7 ) 2 2 3x −5x 72x−3
• Example: ( 2x − 3) ( 3x − 5x + 7 ) 2 2 3x −5x 72x 3 2 6x −10x 14x−3 −9x 2 15x −27
• Example: ( 2x − 3) ( 3x − 5x + 7 ) 2 2 3x −5x 72x 3 2 6x −10x 14x−3 −9x 2 15x −27 3 2 = 6x − 19x + 29x − 27
• Extra HelpHere’s a cool site. Enter the polynomialsyou wish to multiply and it gives you theanswer. A description of how to multiplythe polynomials is included.If the above hyperlink doesn’t work, copy/paste this link: http://www.webmath.com/polymult.html
• FANTASTIC job! You areready to Master theAssignment. Good Luck!