Chapter 5  Polynomials and Polynomial Functions


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1 Math Spring 2009 Chapter 5  Polynomials and Polynomial Functions 5.1 Addition and Subtraction of Polynomials Definition 1. A polynomial is a finite sum of terms in which all variables have whole number exponents (recall whole numbers are positive and include 0) and no variable appears in the denominator. EX The following are not polynomials. State why not: x 1 2, 1 x + 1, x 2 2. Here are a couple examples of polynomials: (a) 7x 6 + 5x 4 2x (b) x 3 y 2 2x 2 y + 9xyz 5 Recall that the degree of a term is the sum of the exponents of the variables. The leading term of a polynomial is the term of highest degree and the leading coefficient is the coefficient of the leading term. EX 2. For the polynomial 7x 6 + 5x 4 2x we have: Degree of the polynomial: 6 Leading Term: 7x 6 Leading Coefficient: 7 Classifying Polynomials We can classify polynomials in a couple of ways. First by the number of terms: Terminology Number of terms Examples Monomial one term 4x 2, 7x 3 y 2, 6 Binomial two terms x 1, x 3 y Trinomial three terms x 2 + 2x 1, x 3 y + 2y x We can also categorize polynomials by the degree of the polynomials Linear polynomial  polynomial of degree 0 or 1. Quadratic Polynomial  polynomial of degree 2. Cubic Polynomial  polynomial of degree 3. After degree three we usually just refer to the polynomial as nth degree, ie 4th degree, 5th degree, etc. We generally will write polnomials in descending order in terms of degree. EX 3. For example we would write x + 6x 2 + 2x 3 1 as 2x 3 + 6x 2 + x 1 Where we start with the highest degree term and continue in descending order. 1
2 5.1.1 Polynomial Functions Polynomials can also be written as function (using function notation). EX 4. The following are polynomial functions. f(x) = x 2 + 3x + 2 P (x) = 2x 3 3x 2 + x 7 When written as a function, we can graph polynomials. EX Graph y = x 2 The graph we get is called a parabola. All quadratic polynomials have parabolas as graphs. In this case the parabola opens upwards. Whenever the leading coefficient is positive the parabola opens upwards. 2. Graph y = x 2 We notice that we now have a negative leading coefficient and the parabola opens downwards Add and Subtract Polynomials To add or subtract polynomials we just combine like terms EX 6. Simplify the following: 1. Add (3x 2 2x + 7) + (2x 2 + x 3) 2. Add (2x 2 y 4xy + 7) + (3x 2 y + 5xy 2y 8) 2
3 3. Subtract ( x 2 + 5x 13) from (x 3 2x + 9) 4. Simplify 8x 2 y 7xy (x 2 y 2xy 2 + 6y) 5.2 Multiplication of Polynomials We learn how to multiply polynomials in steps Multiply a Polynomial by a Monomial To begine we recall product rule for exponents a m a n = a m+n EX 7. Monomial Monomial: Multiply (5x 2 y)(6x 8 y 2 ) In order to do more complicated products we need to recall the distributive property a(b + c + d + + n) = a b + a b + a c + a d + + a n EX 8. Multiply: 3x 2 y(2xy + 7xy 2 8) Multiply a Binomial by a Binomial Abstractly we multiply binomial times a binomial as follows: (a + b)(c + d) = (a + b) c + (a + b) d = a c + b c + a d + b d However we will generally use the FOIL method. FOIL stands for: some examples EX 9. Multiply: 1. (4x 3)(x + 2) F  First O  Outer I  Inner L  Last Let s look at 3
4 2. (3x 2 + 4)(2x 5) Multiply a Polynomial by a Polynomial Let s look at a couple examples. EX 10. Multiply: 1. Multiply x x by 4x Multiply 2x 2 + 5xy 4y 2 by x + 3y Some Formulas and Special Cases The following situations come up often enough that we look at them seperately: (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 (a b) 2 = (a b)(a b) = a 2 2ab + b 2 and (a + b)(a b) = a 2 b 2 EX 11. Multiply: 1. (x + 2) 2 (warning: (x + 2) 2 x 2 + 4) 2. (3x 2 7y) 2 3. [x + (y 3)] 2 4
5 4. (4x )(4x 2 5 ) 5. (7x + y 5 )(7x y 5 ) 5.3 Division of Polynomials Division by Monomials Recall the following properties that we will use: and the quotient rule for exponents EX 12. Divide: 1. 12x 5 y 8 3x 2 y 6 a + b c a m a n = a c + b c = am n 2. 11r 7 s 1 0 4rs x 2 15x+8 3x Divide a Polynomial by a Binomial To divide two polynomials we use a procedure very similar to long division. Let s look at examples: EX 13. Divide: 1. x 2 +11x+28 x x 2 13x+2 2x 5 5
6 3. 6x 2 16x+4x x Factoring Recall, in section 5.2 we learned how to multiply two polynomials together. In this section we will learn how to undo that multiplication. When we do this, we will say we are factoring a polynomial. Factoring is the opposite of multiplying Find the Greatest Common Factor We will first learn to factor a monomial (single term) from a polynomial. The greatest common factor (GCF) is the product of the factors common to all terms in the polynomial. EX Consider the terms 4x 4 y 2 z, 6x 3 y 4, 9xz 8. Find the greatest common factor (GCF) among the terms. 2. Find the GCF: 28(x 4) 2, 7(x 4) 5, 14(x 4) 3 Now that we know how to find the GCF, let s factor a monomial from a polynomial: Steps to factoring out the GCF 1. Determine the GCF of all terms in the polynomial. 2. Write each term as the product of the GCF and another factor. 3. Use the distributive property to factor out the GCF. EX Factor 6x 5 4x x 3 6
7 2. Factor 33x 4 y + 15x 3 y 2 6x 2 y 3 3. Factor 3b 3 + 6b 2 30b 4. Factor 9x(2x 3) + 7(2x 3) 5. Factor 15(3x + 7) + 5(3x + 7) 6. Factor (5x + 1)(m + n) (2x 1)(m + n) Factor by Grouping When a polynomial contains four terms, it may be possible to factor by grouping. Let s see how: EX Factor mx + my + nx + ny 2. Factor x 3 + 6x 2 4x 24 7
8 3. Factor x 3 4x+6x 2 24 (notice this is the same as above but with the middle terms switched) 5.5 Factoring Trinomials Recall that a trinomial is a polynomial with three terms. We will specifically factor polynomials of the form: ax 2 + bx + c We will look at a couple methods for doing this. First, if a = 1 we can use a guess and check method: EX Factor x 2 x Factor m 2 15m + 56 A more systematic method is to factor trinomials by grouping. Let s look at the steps for this: If our polynomial is of the form ax 2 + bx + c To factor by grouping: 1. Find two numbers whose product is a c and whose sum is b. 2. Rewrite the middle term, bx, using the numbers found in step Factor by grouping. EX 18. Factor p 2 3pq 18q 2 WARNING: The fisrt step when factoring any trinomial is to determine whether all three terms have a common factor. If they do, you must first factor out the common factor. EX
9 1. Factor 2x 5 8x 4 42x 3 2. Factor 4x 2 13x Factor 10x 2 7x Factoring Trinomials Using Substitution Sometimes we can use our methods developed above to factor more complicated polynomials. EX Factor y 4 + y Factor 5z 6 13z Factor 2(x + 5) 2 5(x + 5) 12 9
10 5.6 Special Formulas We look at several important and common forms of polynomials and formulas to factor them Difference of Squares The Formula: a 2 b 2 = (a + b)(a b) Check this formula by foiling the right hand side. EX Factor 81m 2 4n 2 2. Factor 3x 4 48y 4 3. Factor (x 7) 2 25 Warning: We can NOT factor polynomials of the form a 2 + b Perfect Square Trinomials The Formulas: a 2 + 2ab + b 2 = (a + b) 2 a 2 2ab + b 2 = (a b) 2 EX Factor x 2 10x
11 2. Factor 16m 4 24m Facotr 9a ab + 25b The Sum and Difference of Two Cubes The Formulas: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b 2 ) EX Factor x Factor 64a 3 27b 6 3. Factor 5x 3 40y Review of Factoring See chart on page 354 and handout. 5.8 Polynomial Equations A polynomial equation is an equation where two polynomials are set equal to each other. 11
12 EX 24. The following are examples of polynomial equations: x 3 + 3x = 2x 5 5x 2 + 3x 2 = 0 We will primarily focus on quadratic equations, ones with highest degree two. For these we will write them in standard form: ax 2 + bx + c = Zero Factor Property For all real numbers a and b, if a b = 0 then either a = 0 or b = 0, or both a and b are zero. In other words, if the product of two numbers is zero, then one or both of the numbers is zero. Why is this useful? EX 25. Use the zero factor property to solve the equation Use Factoring to Solve Equations Steps to solving quadratic equations: 1. Write the equation in standard form. 2. Combine like terms and factor (x + 6)(x 1) = 0 3. Set each factor (with a variable) equal to zero and solve each equation. 4. Check the solution in the original equation. EX Solve 7x 2 = 56x 2. Solve (5x 1)(x + 2) = 6x 3. Solve 3x 2 + 7x 24 = x 12
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