(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES • Be able to identify polynomial, rational, and algebraic expressions. • Understand terminology and notation for polynomials. PART A: DISCUSSION • In Chapters 1 and 2, we will discuss polynomial, rational, and algebraic functions, as well as their graphs. PART B: POLYNOMIALS Let n be a nonnegative integer. An nth -degree polynomial in x , written in descending powers of x, has the following general form: n n 1 an x + an 1x + ... + a1x + a0 , ()an 0 The coefficients, denoted by a , a ,…, a , are typically assumed to be real 1 2 n numbers, though some theorems will require integers or rational numbers. an , the leading coefficient, must be nonzero, although any of the other coefficients could be zero (i.e., their corresponding terms could be “missing”). n an x is the leading term. a is the constant term. It can be thought of as a x0 , where x0 = 1. 0 0 • Because n is a nonnegative integer, all of the exponents on x indicated above must be nonnegative integers, as well. Each exponent is the degree of its corresponding term. (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.2 Example 1 (A Polynomial) 5 4x3 x2 + 1 is a 3rd-degree polynomial in x with leading coefficient 4, 2 leading term 4x3 , and constant term 1. The same would be true even if the terms were reordered: 5 1 x2 + 4x3 . 2 5 The polynomial 4x3 x2 + 1 fits the form 2 n n 1 an x + an 1x + ... + a1x + a0 , with degree n = 3. It can be rewritten as: 5 4x3 x2 + 0x + 1, which fits the form 2 3 2 a3 x + a2 x + a1x + a0 , where the coefficients are: a 4 leading coefficient 3 = () 5 a = 2 2 a = 0 1 a = 1 constant term 0 () § Example Set 2 (Constant Polynomials) th 0 7 is a 0 -degree polynomial. It can be thought of as 7x . 0 is a polynomial with no degree. § (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.3 PART C: CATEGORIZING POLYNOMIALS BY DEGREE Degree Type Examples 0 [Nonzero] Constant 7 1 Linear 3x + 4 2 Quadratic 5x2 x + 1 3 Cubic x3 + 4x 4 Quartic x4 5 Quintic 5 x PART D: CATEGORIZING POLYNOMIALS BY NUMBER OF TERMS Number Type Examples of Terms 1 Monomial x5 2 Binomial x3 + 4x 3 Trinomial 5x2 x + 1 PART E: SQUARING BINOMIALS Formulas for Squaring Binomials 2 a + b = a2 + 2ab + b2 () 2 a b = a2 2ab + b2 () WARNING 1: When squaring binomials, don’t forget the “middle term” of the resulting Perfect Square Trinomial (PST). 2 For example, ()x + 3 = x2 + 6x + 9. Observe that 6x is twice the product of the terms x and 3: 6x = 2 x 3 . ()() 2 2 2 The figure below implies that ()x + y = x + 2xy + y for x > 0 and y > 0 . (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.4 PART F: RATIONAL AND ALGEBRAIC EXPRESSIONS A rational expression in x can be expressed in the form: polynomial in x nonzero polynomial in x Example Set 3 (Rational Expressions) Examples of rational expressions include: 1 a) . x 5x3 1 b) . Irrational coefficients such as 2 are permissible 2 x + 7x 2 as coefficients of either polynomial. x7 + x c) x7 + x which equals . In fact, all polynomials are rational 1 expressions. § An algebraic expression in x is also permitted to contain non-integer rational powers of variable expressions (and their equivalents in radical form). Example Set 4 (Algebraic Expressions) Examples of algebraic expressions include: 1/2 a) x , or x . x3 + 7x5/7 b) . x 3 x + 5 + • (See Footnote 1.) All rational expressions are algebraic. § (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.5 The Venn diagram below is for expressions in x that correspond to functions (see Chapter 1): FOOTNOTES 1. Algebraic expressions. Some sources forbid the presence of in an algebraic expression, since is a transcendental (i.e., non-algebraic) number. That means that is not a zero of any polynomial with integer coefficients, as, say, 2 is. (Section 0.7: Factoring Polynomials) 0.7.1 SECTION 0.7: FACTORING POLYNOMIALS LEARNING OBJECTIVES • Know techniques and formulas for factoring polynomials. • Know the Test for Factorability for factoring quadratic trinomials. • Recognize polynomials in quadratic form and be able to factor them. PART A: DISCUSSION • Factoring is a very commonly used technique in precalculus and calculus. Factoring helps us simplify expressions, find zeros, solve equations and inequalities, and find partial fraction decompositions (see Section 7.3). • Rewriting a sum of terms as a product of factors helps us perform sign analyses, as we will see in Sections 2.4 and 2.10. PART B: FACTORING OUT GCFs For now, when we factor a polynomial, we factor it completely over the integers ( ), meaning that the factors cannot be broken down further using only integer coefficients. That is, the factors must be prime (or irreducible) over the integers. • In Chapter 2, we will factor over other sets, such as , , or . TIP 1: The Greatest Common Factor (GCF), if it is not 1, should typically be factored out first, although it can be factored out piece-by-piece for more complicated expressions. (Unfortunately, there is no simple, standard definition for the GCF.) Example 1 (Factoring out a GCF) We factor 8x + 6 as 24()x + 3 , because 2 is the GCF. 2 is the greatest common divisor of 8 and 6. § Example 2 (Factoring out a GCF) We factor x5 + x3 as x3 ()x2 + 1 , because x3 is the GCF. x3 is the power of x with the least exponent. § (Section 0.7: Factoring Polynomials) 0.7.2 TIP 2: Sometimes, it is helpful to factor out 1, particularly when a polynomial has a negative leading coefficient. Example 3 (Factoring out -1 First) Factor 8x5 6x3 . § Solution 8x5 6x3 = 8x5 + 6x3 () = 2x3 4x2 + 3 () WARNING 1: Sometimes, people confuse signs if they try to 3 factor out 2x immediately. § WARNING 2: Be careful when factoring the base of a power. Make sure to apply the exponent to all factors. Example 4 (Factoring out of a Power) 5 5 ()x3 + x is not equivalent to xx()2 + 1 . The following is correct: 5 5 3 2 ()x + x = xx()+ 1 5 = x5 ()x2 + 1 Each factor of the base must be raised to the exponent, 5. n See Section 0.5, Law 4: xy = xn yn . § () (Section 0.7: Factoring Polynomials) 0.7.3 PART C: FACTORING FORMULAS Factoring Formulas Factoring a … Formula 2 2 2 Perfect Square Trinomial a + 2ab + b = ()a + b (PST) 2 2 2 a 2ab + b = a b () (A rule will be provided for a2 + b2 Sum of Two Squares when we discuss imaginary numbers in Section 2.1. As is, it is prime for now.) Difference of Two Squares a2 b2 = a + b a b ()() Sum of Two Cubes a3 + b3 = ()a + b ()a2 ab + b2 Difference of Two Cubes a3 b3 = a b a2 + ab + b2 ()() WARNING 3: Many math students forgot or never learned the last two formulas. WARNING 4: In the last two formulas, there is no “2” or “ 2” coefficient on the ab term of the trinomial factor. If a and b have no common factors (aside from 1 and 1), the trinomial factors are typically prime. Sometimes, people confuse these trinomials with Perfect Square Trinomials (PSTs), which we introduced in Section 0.6, Part E. TIP 3: In the last two formulas, observe that the binomial factor is “as expected”: ()a + b for a3 + b3 , and ()a b for a3 b3 . The visible signs on the right-hand sides follow the pattern: “same,” “different,” and “+.” (Section 0.7: Factoring Polynomials) 0.7.4 PART D: TEST FOR FACTORABILITY and PRACTICE EXAMPLES Test for Factorability This test applies to any quadratic trinomial of the form ax2 + bx + c , where a, b, and c are nonzero, integer coefficients. (Assume the GCF is 1 or 1; if it is not, factor it out.) The discriminant of the trinomial is b2 4ac . • If the discriminant is a perfect square (such as 0, 1, 4, 9, etc.; these are squares of integers), then the trinomial can be factored over the integers. For example, x2 + 3x + 2 has discriminant 1 and can be factored as x + 2 x + 1 . ()() •• In fact, if the discriminant is 0, then the trinomial is a perfect square trinomial (PST) and can be factored as the square of a 2 binomial with integer coefficients. For example, x 6x + 9 has 2 discriminant 0 and can be factored as ()x 3 . • If the discriminant is not a perfect square, then the trinomial is prime over the integers. This test may be applied in Example Set 5, a) through g), which serve as review exercises for the reader. • The discriminant is denoted by (uppercase delta), though that symbol is also used for other purposes. It is seen in the Quadratic Formula in Section 0.11. We will discuss a method for factoring quadratic trinomials using the Quadratic Formula in Chapter 2. (Section 0.7: Factoring Polynomials) 0.7.5 Example Set 5 (Factoring Polynomials) Factor the following polynomials over the integers. 2 a) x + 9x + 20 2 b) x 20x + 100 (Hint: This is a Perfect Square Trinomial (PST).) 2 c) x 4x 12 d) 3x2 20x 7 2 e) 4x + 11x + 6 2 f) 2x + 10x + 5 g) 3x2 6x 3 + 4 h) x 16 i) a3 3a + 2a2b 6b (Hint: Use Factoring by Grouping.