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# Polynomials

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First Quarter - Chapter 1

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### Polynomials

1. 1. Let’s Define it.
2. 2. P O LY N O M I A L V O C A B U L A R YTerm – a number or a product of a number and variables raised to powersCoefficient – numerical factor of a termConstant – term which is only a numberPolynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.
3. 3. P O LY N O M I A L V O C A B U L A R YIn the polynomial 7x5 + x2y2 – 4xy + 7 There are 4 terms: 7x5, x2y2, -4xy and 7. The coefficient of term 7x5 is 7, of term x2y2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term.
4. 4. T Y P E S O F P O LY N O M I A L SMonomial is a polynomial with 1 term.Binomial is a polynomial with 2 terms.Trinomial is a polynomial with 3 terms.Multinomial is a polynomial with 4 or more terms.
5. 5. DEGREESDegree of a term To find the degree, take the sum of the exponents on the variables contained in the term. Degree of a constant is 0. Degree of the term 5a4b3c is 8 (remember that c can be written as c1).Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial. Degree of 9x3 – 4x2 + 7 is 3.
6. 6. E VA L U AT I N G P O LY N O M I A L SEvaluating a polynomial for a particular value involvesreplacing the value for the variable(s) involved. ExampleFind the value of 2x3 – 3x + 4 when x = 2. 2x3 – 3x + 4 = 2( 2)3 – 3( 2) + 4 = 2( 8) + 6 + 4 = 6
7. 7. COMBINING LIKE TERMSLike terms are terms that contain exactly the same variables raisedto exactly the same powers. Warning! Only like terms can be combined through addition and subtraction.ExampleCombine like terms to simplify. x2y + xy – y + 10x2y – 2y + xy = x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together) = (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y
9. 9. ADDING AND SUBTRACTING POLYNOMIALSAdding Polynomials Combine all the like terms.Subtracting Polynomials Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms.
10. 10. ADDING AND SUBTRACTING POLYNOMIALSExampleAdd or subtract each of the following, as indicated.1) (3x – 8) + (4x2 – 3x +3) = 3x – 8 + 4x2 – 3x + 3 = 4x2 + 3x – 3x – 8 + 3 = 4x2 – 52) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 83) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7) = – a2 + 1 – a2 + 3 + 5a2 – 6a + 7 = – a2 – a2 + 5a2 – 6a + 1 + 3 + 7 = 3a2 – 6a + 11
11. 11. ADDING AND SUBTRACTING POLYNOMIALSIn the previous examples, after discarding theparentheses, we would rearrange the terms sothat like terms were next to each other in theexpression.You can also use a vertical format in arrangingyour problem, so that like terms are aligned witheach other vertically.
12. 12. MULTIPLYING POLYNOMIALS Let’s Multiply!
13. 13. M U LT I P LY I N G P O LY N O M I A L SMultiplying polynomials • If all of the polynomials are monomials, use the associative and commutative properties. • If any of the polynomials are not monomials, use the distributive property before the associative and commutative properties. Then combine like terms.
14. 14. Multiplying Polynomials ExampleMultiply each of the following.1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x32) (4x2)(3x2 – 2x + 5) = (4x2)(3x2) – (4x2)(2x) + (4x2)(5) (Distributive property) = 12x4 – 8x3 + 20x2 (Multiply the monomials)3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5) = 14x2 + 10x – 28x – 20 = 14x2 – 18x – 20
15. 15. Multiplying Polynomials Example Multiply (3x + 4)2 Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4).(3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4) = 9x2 + 12x + 12x + 16 = 9x2 + 24x + 16
16. 16. Multiplying Polynomials Example Multiply (a + 2)(a3 – 3a2 + 7).(a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7) = a4 – 3a3 + 7a + 2a3 – 6a2 + 14a4 – a3 – 6a2 + 7a + 14 =
17. 17. Multiplying PolynomialsExampleMultiply (3x – 7y)(7x + 2y) (3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y) = 21x2 + 6xy – 49xy + 14y2 = 21x2 – 43xy + 14y2
18. 18. Multiplying Polynomials ExampleMultiply (5x – 2z)2(5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z) = 25x2 – 10xz – 10xz + 4z2 = 25x2 – 20xz + 4z2
19. 19. Multiplying PolynomialsExampleMultiply (2x2 + x – 1)(x2 + 3x + 4)(2x2 + x – 1)(x2 + 3x + 4) = (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4) = 2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4 = 2x4 + 7x3 + 10x2 + x – 4
20. 20. SPECIAL PRODUCTS Let’s multiply!
21. 21. THE FOIL METHODWhen multiplying 2 binomials, the distributiveproperty can be easily remembered as the FOILmethod. F – product of First terms O – product of Outside terms I – product of Inside terms L – product of Last terms
22. 22. Using the FOIL Method ExampleMultiply (y – 12)(y + 4) (y – 12)(y + 4) Product of First terms is y2 (y – 12)(y + 4) Product of Outside terms is 4y (y – 12)(y + 4) Product of Inside terms is -12y (y – 12)(y + 4) Product of Last terms is -48 F O I L (y – 12)(y + 4) = y2 + 4y – 12y – 48 = y2 – 8y – 48
23. 23. Using the FOIL MethodExample Multiply (2x – 4)(7x + 5) F L F O I L (2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5) I O = 14x2 + 10x – 28x – 20 = 14x2 – 18x – 20We multiplied these same two binomials together in theprevious section, using a different technique, but arrived at thesame product.
24. 24. Special ProductsIn the process of using the FOIL method on products ofcertain types of binomials, we see specific patterns thatlead to special products.Squaring a Binomial (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2Multiplying the Sum and Difference of Two Terms (a + b)(a – b) = a2 – b2
25. 25. Special ProductsAlthough you will arrive at the sameresults for the special products by usingthe techniques of this section or lastsection, memorizing these products cansave you some time in multiplyingpolynomials.
26. 26. DIVIDING POLYNOMIALS Let’s divide!
27. 27. D I V I D I N G P O LY N O M I A L SDividing a polynomial by a monomial Divide each term of the polynomial separately by the monomial. 3 3 12 a 36 a 15 12 a 36 a 15Example 3a 3a 3a 3a 2 5 4a 12 a
28. 28. DIVIDING POLYNOMIALS D I V I D I N G P O LY N O M I A L S Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers, which is reviewed on the next slide.
29. 29. DIVIDING POLYNOMIALS D I V I D I N G P O LY N O M I A L S 168 Divide 43 into 72. Multiply 1 times 43.43 7256 Subtract 43 from 72. 43 Bring down 5. 29 5 Divide 43 into 295. 258 Multiply 6 times 43. Subtract 258 from 295. 37 6 Bring down 6. 344 Divide 43 into 376. 32 Multiply 8 times 43. Subtract 344 from 376. We then write our result as 32 Nothing to bring down. 168 . 43
30. 30. Dividing P O L Y N O M I A L S DIVIDING PolynomialsAs you can see from the previous example, there isa pattern in the long division technique. Divide Multiply Subtract Bring down Then repeat these steps until you can’t bring down or divide any longer.We will incorporate this same repeated techniquewith dividing polynomials.
31. 31. DIVIDING POLYNOMIALS D I V I D I N G P O LY N O M I A L S 4x 5 Divide 7x into 28x2. Multiply 4x times 7x+3. 27x 3 28 x 23 x 15 Subtract 28x2 + 12x from 28x2 – 23x. 2 Bring down – 15. 28 x 12 x Divide 7x into –35x. 35 x 15 Multiply – 5 times 7x+3. Subtract –35x–15 from –35x–15. 35 x 15 Nothing to bring down. So our answer is 4x – 5.
32. 32. Dividing P O L Y N O M I A L S DIVIDING Polynomials 2 x 10 Divide 2x into 4x2. 2 Multiply 2x times 2x+7.2 x 7 4x 6x 8 Subtract 4x2 + 14x from 4x2 – 6x. 2 4 x 14 x Bring down 8. 20 x 8 Divide 2x into –20x. 20 x 70 Multiply -10 times 2x+7. Subtract –20x–70 from –20x+8. 78 Nothing to bring down.We write our final answer as 2 x 10 78 ( 2 x 7)
33. 33. THE ENDGOODBYE! 