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  • 1.Advanced Algebra1 This chapter aims the following objectives: 1. To generalize Polynomial.2. To familiarize the operation used in polynomial3. To restate Synthetic Division4. To determine the zeros of Polynomial5. To derive Quadratic Formula

2. Advanced Algebra 2LESSON 1: POLYNOMIALSObjectivesTo define polynomials. To be aware of the terms of a polynomial. To distinguish the degree of a polynomial. Hi Everybody! I want to introduce to you my friend, Polynomials. He is onevariable of an algebraic expression of the formanxn + an-1xn-1 + an-2xn-2 + + a0, where n is a non-negative integer, and an, an-1, an-2, ,a0 are constants, and an 0.You want to know him better? Here is an example of a polynomial. x2 + 4xy + 4y2 3. Advanced Algebra 3After you have learned what polynomial is, you should also know his different terms and degrees.I want you to meet Monomial. He is a polynomial consisting of only one term.Another is Binomial. He is a polynomial consisting of two terms.And the last but not the least is a Trinomial. He is a polynomial consisting of three terms.Now you meet them, I want you to know more about them because it will help you a lot. Here are the examples.Hi there. I ammonomial 5x; -2ab; 11a3; 28And I am binomial3x2 + y; 8x2y2 4; ; a2 + 2And we are the trinomials. 6x2 + 2xy + y2; m + n p; x2 + 5x + 6Yes! This is it. You can now easily determine the terms of a polynomials. Youre now ready to take my challenges. But wait youre in the half of our study were not yet done. You must know what the degree of polynomial is. 4. Advanced Algebra 4 Remember that in determining the degree of a polynomial you must get the highestpower of its terms after it has been simplified. 5 is the degree of polynomial becauseon the term 6x2y x is in the highestpower, then x is in the 2nd degree andon the term 12x3y2, x is in the 3rd degree so we add same variables exponent then we will get 5 as the degree of the polynomial. 6x2y + 12x3y2 + 8 And now, this is the moment of truth youre now ready to solve and answer all my challenges regarding our lesson which is polynomials. Im sure that youll get excellent points if you really understand our lesson. 5. Advanced Algebra5NAME: RATING:A. Determine whether the following is a polynomial or not. 1. 3x2 5x + 6 ______ 2. A2b2 4ab + 8 ______ 8. q + c + c2______ 3. x2 y 2 ______ 9. ______4. a3 + b3 + c3 ______ 10. 5x +______5.______ 11.______6.______ 12.______7. x1/3 + 5 ______ B. Classify each polynomial according to the number of its terms. 1. 8a + 2 ______6. 5x2y 12xy + y2 ______ 2. -5xy + x + xy______ 7.xyz______ 3. +x ______ 8. m + 2n 3p ______ 4. a2 + 4a + 16 ______ 9.______5.x + y3______ 10. 2a3 5a2 15______ C. Give the degree of each of the following polynomials.1. 4x______5. 3a4b2 + 4ab3 6b7 ______ 2. 4x2 + 4x 8______6. a3b4 + a2b2 6ab3______ 3. 5xy + 8y2 + 13x2______7. 9 6xy + yz + xyz4. 6x2 + 4x +____________ 6. Advanced Algebra68. v- 9. 8xyz2 - ____________ 10. m + 9m5n8 + 3m3n5p7 - 7np10______LESSON 2:FUNDAMENTAL OPERATIONS Objectives To explore the addition with subtraction operation.To familiarize student the use of multiplication and division.To perform the synthetic division method. After we discussed the polynomials terms and degree, I will now introduce to you thedifferent operation in solving polynomial expression. Meet addition and subtraction, and follow theirrules in solving polynomials. 7. Advanced Algebra 7Remember that in adding polynomials, simply add the numerical coefficient of liketerms. And in subtracting polynomials, just simply change the second polynomialinto its additive inverse and then proceed to addition.Here is an example of adding polynomials. (9x4 x3 + 5x2 8x) + (4x4 x3 7x + 7) =?Solution: (9x4 x3 + 5x2 8x) + (4x4 x3 7x + 7) = 9x4 x3 + 5x2 8x + 4x4 x3 7x + 7 = 9x4+ 4x4 x3 x3+ 5x2 8x 7x + 7 = 13x4 2x3 + 5x2 15x + 7 And here is the example of subtracting polynomials.(15x2 4xy + 10y2) (9x2 + 4xy + 5y2) =? Solution:(15x2 4xy + 10y2) (9x2 + 4xy + 5y2) = 15x2 4xy + 10y2 9x2 4xy 5y2 = 15x2 9x2 4xy 4xy + 10y2 5y2 = 6x2 8xy + 5y 8. Advanced Algebra 8 NAME:RATING:Perform the indicated operations.1. (6xy 2x +4) + (11 8xy + 6x)2. (-8mn2 + 12n) (m3 4m2 + 4n)3. (3b2 2b + 9) + (b2 + 6b 4)4. (5x2 2y + 4z) (x2 3y2 + z)5. (a2 + 4a + 2) + (2a + 3)6. (3x2 y) (2x2 + 5y)7. (-7x4y3 21x3y3 + 28x5y4) + (7x2y2 + 6xy + 2x2y)8. (-x2 + 6x 2) (x2 x + 3) (x + 1) + (x + 2)9. (3y2 + 3xy + 10) + (4y3 10xy 15)10. (-2m2 + mn + 5n2) (-4m2 6mn + 3n2) After we have discussed the two operations which are addition and subtraction, we now proceed to Multiplication and division, the other operation in polynomials. You will meet them later in the middle of our study. We all know that mathematics have 4 major operations in solving an equation. Like in arithmetic calculation, polynomials have also those 4 operations. And we formerly discussed addition and subtraction. Now I will introduce to you multiplication. 9. Advanced Algebra9Remember that to get the product of polynomial; you must multiply each term of polynomial by each of the other terms of polynomial. Then combine like terms.Here is the example of multiplication for you to understand what multiplication meant to be.(4x + 3) (2x + 5)Solution:(4x + 3) (2x + 5) = ( 4x)(2x) + (4x)(5) + (3)(2x) + (3)(5) = (8x2 + 20x) + (6x + 15) = 8x2 + (20x + 6x) + 15 = 8x2 + 26x + 15 Rules in Dividing Polynomials1) In dividing polynomials by monomial, divide each term of a polynomial bymonomial.2) In dividing polynomial by another polynomial:a. Arrange the term in descending power with respect to a variable.b. Get the common factor of the dividend and divisor. 10. Advanced Algebra10 To make it clear, here is the example that clarifies to your curiosity.We factor the dividend with the common factor of the divisor. Solution: = Then I cancel it out = = (a + 2) And this is it. Now you answer again my challenges for me to know that your ability in learning my lesson is satisfy your knowledge. NAME:RATING:A. Perform the indicated operations.1. (c2 16)(c +1)2. (11p2 66p + 99)(p + 1)3. (a b)2(a + b)2 11. Advanced Algebra 11 4. (5x2y + 3xy2 7x2y3)(xy)5. (a2 + 2a + 3)(a 5) +6. (x + 1)(x + 2)(x + 3)7. (x2 + x + 2)(2x2 + 3) + 8.9.10. (a2n 3an + 5)(a + 2)(a2 + 4a + 2) B. Find the Quotient.1. (a2 7a + 10) (a 5) 2. (x3 4x2 2 + 5x)(x 1) 3. (3a4 2a + 5) (a2 + 3) 4. (2x3 + 5x2 x 1) (x 1) 5. (2x2 5x 6)(2x 1) LESSON 3: SYNTHETIC DIVISION To define Synthetic division. Objectives of the procedure in using Synthetic division. To be aware To perform synthetic division in solving polynomial function. 12. Advanced Algebra12 I know that you have a difficulty in getting the common factor to divide the polynomials. I have here a friend that will help you to make your factoring in its easiest and shortest way. I proudly introduce to you Synthetic division. He is a process of division for polynomials in one variable where the divisor is of the form x c, and c is any real number. So your difficulty in dividing polynomials will lessen through this method. I think that you will use this in the near future of your study in different branches of mathematics. 13. Advanced Algebra 13 Procedure in using synthetic division1. Arrange the terms into descending power.2. Copy the numerical coefficient. (If the descending power of the terms is like this x3+ 2x + 1 then the arrangement should be10 21: becauseththe degree of polynomials is in the 4 . )3. Substitute the value of x in the divisor.4. Then, bring down 1st the numerical coefficient w/c is in his 1st term.5. Multiply to the value of x then subtract product to the 2nd term of numericalHere is the example for you to apply this method.(x3 + 9x2 + 17x 19)(x + 4)Solution:-4 1917 -19-4 -2012remainder15-3 -7 The final answer = x2 + 5x 3 So, we finally studied this method. And now, you can use it in your mathematics subject. And of course, after we end the lesson we might have a test regarding this lesson. Are you ready? But you should be ready because I rate you according to this scale: NAME: RATING: 14. Advanced Algebra 14Get the quotient of the following by using the Synthetic Division Method.1. (x4 + 4x3 + x2 + x + 17)(x 2)2. (x4 + 2x2 + x - 11) (x + 5)3. (x2 + 4x + 21)(x + 7)4. (2x4 + x3 x 12) (x 2)5. (x3 4x2 + 5x 2) (x 1) 6. (x3 7x2 4x + 24)(x 6)7. (3x3 2x2 + 5x +1) (x + 5)8. x5 2x4 + 3x3 2x2 + 1)(x 2)9. (x2 x 20) (x 5)10. (x3 4x2 2 + 5x)(x 4)LESSON 4: ZEROS OF APOLYNOMIAL FUNCTIONObjectivesTo determine the zeros of a polynomials. To be aware in the different techniques in factoring a polynomials. To perform the use of quadratic formula in solving polynomials. 15. Advanced Algebra15Its a long discussion about the operations used in Polynomials. But now another mathematics word you will be meet. I introduced to you the Zeros of polynomial Function. And from the word itself you will be notice that he is pertaining to zero.Zero of a Polynomial Function is the value of t he variable x, which makes the polynomial function equal to zero. And its written in symbolic form:f(x) = 0 x = -2; x2 + 4x + 4f(x) = (-2)2 + 4(-2) + 4Thats the simplest explanation of it.And I think you already understand it= 4 + (-8) + 4or maybe Ill give an example of that. =88=0 16. Advanced Algebra16Now, take my challenges. You should getperfect score, because this lesson is very easy anyone can perfect it. So grab it! NAME:RATING:Which numbers -3, -2, -1, 1, 2 and 3 are zeros of the following polynomial functions?1. f(x) = x3 + 4x2 + x 62. f(x) = 2x3 3x2 11x + 63. f(x) = x3 +3x 2 x 3 17. Advanced Algebra 174. f(x) = x4 4x3 +6x2 4x + 15. f(x) = x3 2x2 5x + 66. f(x) = x3 x2 9x + 97. f(x) = x3 3x2 5x + 128. f(x) = 5x4 + 2x3 3x 39. f(x) = 12x3 18x2 + 14x + 1710. f(x) = x3 5x2 + 8x 311. f(x) = x2 5x + 712. f(x) = 8x4 + 5x3 + 2x2 6x 113. f(x) = 3x3 8x2 + 7x 6LESSON 5: FACTORING TECHNIQUES Objectives To determine the di