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LESSON 20addition andmultiplication of polynomialsLESSON 20 Addition and Multiplication of PolynomialsBase 10 and Base X - Recall the factors of each of the pieces in base 10. The unitblock (green) is 1 x 1.The 10 bar (blue) is 1 x 10, and the 100 square (red) is 10 x 10. Each of thesepieces may also be expressed in terms of exponents: 1 x 1 1, which is 10; 1 x 10 10, which is 101; 10 x 10 100, which is 102. Below is the number 156 shownwith the blocks and expressed in different ways.1001 x 1001 x 102 50 5 x 10 5 x 101 66x16 x 100In the decimal system, every value is based on 10.The decimal system is referred to as base 10.In algebra, the unit bar is still one by one. The smooth blue piece that snapsinto the back of the 10 bar is one by X, and the smooth red piece that snaps intothe back of the 100 square is X by X. Each of these pieces may also be expressedin terms of exponents: 1 1 1, which is 10, or X0 (which is the same thing sinceboth are equal to one); 1 X X, which is X1; and X X X2.On the next page is the polynomial X2 5X 6 shown with the blocks andexpressed in different ways.ALGEBRAADDITION AND MULTIPLICATION OF POLYNOMIALS - LESSON 2099

X 2X2X X1 X2 5X5 X5 X1 66 16 X0In algebra, every value is based on X.Algebra is arithmetic in base X.Kinds of Polynomials - Polynomial derives from polys (many) and nomen(name), so literally it means “many names.” If a polynomial has three components,it is called a trinomial (tri- meaning “three”). A binomial (bi- meaning “two”) hastwo parts. A monomial (mono- meaning “one”) has one part.In the next few sections, whenever you feel the need to reassure yourself thatyou are on the right track, simply change the equation from base X to base 10, andredo the problem.On the next two pages, operations that you are familiar with, such as additionand multiplication, will now be performed with polynomials in base X insteadof base 10. Take your time and remember the connection with what you alreadyknow. Someone has said, “Algebra is not difficult, just different.”Addition of Polynomials - When adding or subtracting polynomials,remember that “to combine, they must be the same kind.” Units may be added(or subtracted) with other units, Xs with Xs, X2s with X2s, etc. Since we don’tknow what the value is for X, all the addition and subtraction is done in thecoefficients. Read through the following examples for clarity. The gray inserts areused to show -X.Example 1X2 2X 42 X 5X 62X2 7X 10 100LESSON 20 - ADDITION AND MULTIPLICATION OF POLYNOMIALSALGEBRA

Example 2X2 – X – 4 X2 5X 62X2 4X 2 Multiplication of Polynomials - When we multiply two binomials, the resultis a trinomial. I like to use the same format for multiplying a binomial as for multiplying any double-digit number in the decimal system.Let’s look at a problem in the decimal system using expanded notation.Example 3x13x 12 If this were in base X insteadof base 10, it would look like this:10 310 220 6100 30100 50 6X 3xX 22X 6X2 3XX2 5X 6ALGEBRAADDITION AND MULTIPLICATION OF POLYNOMIALS - LESSON 20101

The area, or product, of thisrectangle is X2 5X 6.Do you see it?In this rectangle, we cover up mostof it to reveal the factors, which are(X 3) over and (X 2) up.2XX3The written equivalent of the picturelooks just like double-digit multiplication, which it is.X 3xX 22X 62X 3XX2 5X 6Example 420 310 3xx2313 If this were in base X insteadof base 10, it would look likethis:60 9200 30200 90 92X 3xX 36X 922X 3X2X2 9X 9102LESSON 20 - ADDITION AND MULTIPLICATION OF POLYNOMIALSALGEBRA

The area, or product, of thisrectangle is 2X2 9X 9. Doyou see it?In this rectangle, we cover upmost of it to reveal the factors,which are (2X 3) over and(X 3) up.3X2X3The written equivalent of the picturelooks just like double-digit multiplication, which it is.2X 3xX 36X 922X 3X2X2 9X 9You can also multiply binomials that include 0negative numbers. To show -X, use the gray inserts.The addition identity tells us that X (-X) 0.ALGEBRAX-XADDITION AND MULTIPLICATION OF POLYNOMIALS - LESSON 20103

Example 5X–3xX 22X – 62X – 3XX2 –X 2X–6X-3Here it is in base 10: 710 – 3 12x 10 28420 – 6100 – 30100 – 10 – 6 84Look at the next two examples carefully.Example 6(X – 1)(X – 5)(X – 2)(X 4)104Example 7X–2xX 44X – 82X – 2XX–1xX–5–5X 52X –XX2 2X – 8X2 – 6X 5LESSON 20 - ADDITION AND MULTIPLICATION OF POLYNOMIALSALGEBRA

20Alesson practiceBuild.1. X2 11X 22. X2 6X 83. X2 – 8Build and add.4.X2 6X 3 3X2 7X 95.X2 8 X2 6X 76.2X2 10X 7 2X2 8X 9Build a rectangle and find the area (product).7. (X 1)(X 2) 8. (X 4)(X 3) 9. (X 1)(X 5) ALGEBRA 1 Lesson Practice 20A241

LESSON PRACTICe 20AMultiply.10.2423X 2 X 111.13.X 8 3X 516.4X 2 X 32X 1X 55X 5 X 212.14.X 3 2X 115.3X 2 2X 117.2X 5 X 218.3X 5 3X 1 ALGEBRA 1

20Blesson practiceBuild.1. X2 – 3X – 72. 2X2 – 7X – 33. X2 5X 9Build and add.4.X2 3X 2 X2 7X 125.X2 6X 5 3X2 X 26.5X2 5X 10 2X2 11X 5Build a rectangle and find the area (product).7. (X 4)(X 5) 8. (X 7)(X 3) 9. (X 4)(X 8) ALGEBRA 1 Lesson Practice 20B243

LESSON PRACTICe 20BMultiply.10. 13.X 8 X 316. 2447X 1X 24X 2X 311.3X 7X 612.2X 8 3X 114.2X 1 X 915.3X 5 X 217.5X 2 3X 318.3X 7 4X 2 ALGEBRA 1

20Csystematic reviewBuild and add.1. 3X2 7X 62.2X2 5X 1 X2 2X 3 X2 3X 43.4X2 8X 2 –X2 3X – 1Build a rectangle and find the area (product).4. (X 4)(X 8) 5. (X 5)(X 2) 6. (X 2)(X 6) Multiply.7.8.3X 6 X 210. Write on one line:1 X 42X 5 X 39.4X 5 X 111. Rewrite using positive exponents: X–3196Simplify. Write expressions with exponents on one line.12. 52 x 30 x 5–4 13. A4 A7 14. (52)5 15. (5)12 (53) ? ALGEBRA 1 systematic review 20C245

Systematic review 20C1 X 416.196 17. C–5 x C2 18. The base of a rectangle is X 4, and the height is X 5. What isthe area of the rectangle? (Remember that the area of a rectangleis base times the height.)19. Find the area of the rectangle in #18 if X equals six.20. Take two times the base and height of the rectangle in #18,using the distributive property, and then find the polynomial thatexpresses the new area.246ALGEBRA 1

20Dsystematic reviewBuild and add.1.X2 – 3X – 72. X2 11X 222 2X 4X – 43.X2 – 10X –2 3X –4X 6 –2X –5X 14Build a rectangle and find the area (product).4. (X 2)(X 7) 5. (2X 3)(X 4) 6. (X 1)(X 9) Multiply.7.2X 48.x X 310. Write on one line:1 X43X – 19.x X 42X – 3x X–41 Y 52251 11. Rewrite using positive exponents:4X1 Y 5225ALGEBRA 1 systematic review 20D247

Systematic review 20DSimplify. Write expressions with exponents on one line.12. 37 x 43 x 4–2 13. B5 B1 14. (83)6 15. (2)15 (23)? 1 X41 Y 516.225 17. D–3 x D8 x D–7 18. The base of a rectangle is 2X 4, and the height is X 4.What is the area of the rectangle?19. Find the area of the rectangle in #18 if X equals 10.20. The area of a second rectangle is X2 3X 1. What is thesum of the area of the two rectangles (from #18 and #20)?248ALGEBRA 1

20Esystematic reviewBuild and add.1. X2 3X – 22. 3X2 2X – 1 X2 4X 33. 5X2 4X 7 2X2 – 2X 8 –X2 3X 7Build a rectangle and find the area (product).4. (X 3)(X 3) 5. (2X 4)(X 2) 6. (3X)(X 2) Multiply.7.x2X – 38.X – 210. Write on one line:1 X5xX–19.X – 6x2X 2X–311. Rewrite using positive exponents: Y–2169Simplify. Write expressions with exponents on one line.12. 7–2 x 75 7–2 ALGEBRA 1 systematic review 20E13. A7 B3 249

Systematic review 20ESimplify. Write expressions with exponents on one line.14. (52) 5 15. (5)12 (53)? 16. – 169 17. C0 C–4 D8 D–7 D–3 C3 1 X518. Stephanie’s savings are represented by 3N 4, and Chuck’s are representedby 2N 5. Write an expression representing their combined savings.19. Stephanie and Chuck have each been saving as described in #18 for10 weeks (N), what is the total amount they have saved?20. The base of a rectangle is 2Y 7, and the height is 7Y 5. What is the areaof the rectangle?250ALGEBRA 1

20Hhonors lessonHere are some more problems involving exponents.Follow the directions and answer the questions.1. Suppose that m represents the mass in grams of a substance that halves insize each month. You can find the value for each month simply by dividingthe value for the previous value by two.x (number of months)0m (mass in grams)20012342.What was the mass of the substance when measuring began? (time 0)3.How long will it be until there are 100 grams remaining?4.How long will it be until there are only 50 grams remaining?5.What is the mass of the substance after four months?ALGEBRA 1 HONORS LESSON 20H251

honors lesson 20H6. Make a graph showing the first five months of decrease of the substancedescribed on the previous page.200160m (mass in grams)1501401201008060402001234x (months)In real life, a scientist may wish to find the value of m for a certain number ofxmonths without finding every value in between. In this case, m 200(.5) , where xstands for the number of months. Compare the example to the corresponding valueon your chart.Examplex m 200(.5) . Find the value of m after four months.m 200(.5)4m 200(.0625) 12.5 grams7. Use the equation given above to find the mass of the substanceafter six months.252ALGEBRA 1

20test1. X2 2X 2 is a:I.III.polynomial II.binomial IV.trinomialmonomialA. I and IIB. I and IVC. I onlyD. II onlyE. III only2. X2 3X 22 X 4X 5A.B.C.D.E.X2 7X 72X2 7X 39X 72X2 7X 72X2 – X 73.X2 X 10 X2 – 2X 4A.B.C.D.E.2X2 – X 14X2 – X 14–X 62X2 – 3X – 62X2 X 144.X2 8X 62 X – 3X – 1A.B.C.D.E.ALGEBRA 1 Test 20X2 5X 52X2 – 5X – 5–11X 72X2 11X 72X2 5X 55. X2 – 5X – 2 X2 – 4X – 3A.B.C.D.E.X2 9X 59X 52X2 – X – 1X2 – 9X – 52X2 – 9X – 56. What is the sum of 2X 3and 4X – 5?A. 6X2 – 2B.6X 2C. 6X – 2D. 6X 8E. 2X 27. What is the sum of2X2 – 9X 5 and X2 4X – 1?A. 3X2 5X 4B. 3X2 – 5X 4C. X2 – 5X 4D. 3X2 13X 4E. 3X2 – 5X 68. 4X 3x X 1A.B.C.D.E.5X2 5X 411X 34X2 7X 34X2 7X 44X2 X 351

test 209. X 3x X 2A.B.C.D.E.X2 6X 5X2 5X 62X2 5X 6X2 X 5X2 X 610. The product of X 4 and X – 2 is:2A. X 2X – 8B.C.D.E.X2 – 2X – 82X2 6X – 8X2 – 6X – 8X2 – 2X 811.Multiply X 1 and X 5.A. X2 5X 6B. X2 6X – 5C. X2 6X 5D. X2 5X 4E. 2X2 6X 513. If 7X 1 and X 2 aremultiplied, the first termof the answer would be:A. X2B. 7X2C. 14X2D. 2X2E. 7X14. If 2X 4 and X 5 aremultiplied, the first termof the answer would be:A. 3X2B. 2X2C. 10X2D. 8X2E. 20X215. When we multiply 2binomials, the result is a(n):A. binomialB. trinomialC. monomialD. integerE. inequality12.Multiply X – 3 and X – 6.A. X2 9X – 18B. X2 9X 18C. 2X2 – 9X 18D. X2 – 9X 18E. X2 – 18X – 952ALGEBRA 1