**Roots & Factors**Roots of a polynomial A root of a polynomial p(x) is a number ↵ 2 R such that p(↵) = 0. Examples. • 3 is a root of the polynomial p(x) = 2x ? 6 because p(3) = 2(3) ? 6 = 6 ? 6 = 0 • 1 is a root of the polynomial q(x) = 15x2? 7x ? 8 since q(1) = 15(1)2? 7(1) ? 8 = 15 ? 7 ? 8 = 0 • ( Be aware: What we call a root is what others call a “real root”, to emphasize that it is both a root and a real number. Since the only numbers we will consider in this course are real numbers, clarifying that a root is a “real root” won’t be necessary. 2p2)2? 2 = 0, so 2p2 is a root of x2? 2. Factors A polynomial q(x) is a factor of the polynomial p(x) if there is a third polynomial g(x) such that p(x) = q(x)g(x). Example. 3x3? x2+ 12x ? 4 = (3x ? 1)(x2+ 4), so 3x ? 1 is a factor of 3x3?x2+12x?4. The polynomial x2+4 is also a factor of 3x3?x2+12x?4. Factors and division If you divide a polynomial p(x) by another polynomial q(x), and there is no remainder, then q(x) is a factor of p(x). That’s because if there’s no remainder, then definition of q(x) being a factor of p(x). If q(x)has a remainder, then q(x) is not a factor of p(x). q(x)is a polynomial, and p(x) = q(x)?p(x) ?. That’s the p(x) q(x) p(x) Example. In the previous chapter we saw that 6x2+ 5x + 1 3x + 1 = 2x + 1 132Multiplying the above equation by 3x + 1 gives 6x2+ 5x + 1 = (3x + 1)(2x + 1) so 3x + 1 is a factor of 6x2+ 5x + 1. * * * * * * * * * * * * * Most important examples of roots Notice that the number ↵ is a root of the linear polynomial x ? ↵ since ↵ ? ↵ = 0. You have to be able to recognize these types of roots when you see them. polynomial root x ? 2 2 x ? 3 3 x ? (?2) ?2 ?2 x + 2 ?15 x + 15 x ? ↵ ↵ Linear factors give roots Suppose there is some number ↵ such that x?↵ is a factor of the polynomial p(x). We’ll see that ↵ must be a root of p(x). That x ? ↵ is a factor of p(x) means there is a polynomial g(x) such that p(x) = (x ? ↵)g(x) Then p(↵) = (↵ ? ↵)g(↵) = 0 · g(↵) = 0 133

Notice that it didn’t matter what polynomial g(x) was, or what number g(↵) was; ↵ is a root of p(x). If x ? ↵ is a factor of p(x), then ↵ is a root of p(x). Examples. • 2 is a root of p(x) = (x ? 2)(⇡7x15? 27x11+3 p(2) = (2 ? 2)(⇡7215? 27(2)11+3 = 0 · (⇡7215? 27(2)11+3 = 0 4x5? x3) because 425? 23) 425? 23) • 4 is a root of q(x) = (x ? 4)(x101? x57? 17x3+ x) • ?2, 1, and 5 are roots of the polynomial 3(x + 2)(x ? 1)(x ? 5). Roots give linear factors Suppose the number ↵ is a root of the polynomial p(x). That means that p(↵) = 0. We’ll see that x ? ↵ must be a factor of p(x). Let’s start by dividing p(x) by (x ? ↵). Remember that when you divide a polynomial by a linear polynomial, the remainder is always a constant. So we’ll get something that looks like p(x) (x ? ↵)= g(x) + c (x ? ↵) where g(x) is a polynomial and c 2 R is a constant. Next we can multiply the previous equation by (x ? ↵) to get p(x) = (x ? ↵) ⇣ ⌘ c g(x) + (x ? ↵) c = (x ? ↵)g(x) + (x ? ↵) = (x ? ↵)g(x) + c (x ? ↵) 134

That means that p(↵) = (↵ ? ↵)g(↵) + c = 0 · g(↵) + c = 0 + c = c Now remember that p(↵) = 0. We haven’t used that information in this problem yet, but we can now: because p(↵) = 0 and p(↵) = c, it must be that c = 0. Therefore, p(x) = (x ? ↵)g(x) + c = (x ? ↵)g(x) That means that x?↵ is a factor of p(x), which is what we wanted to check. If ↵ is a root of p(x), then x ? ↵ is a factor of p(x) Example. It’s easy to see that 1 is a root of p(x) = x3? 1. Therefore, we know that x ? 1 is a factor of p(x). That means that p(x) = (x ? 1)g(x) for some polynomial g(x). To find g(x), divide p(x) by x ? 1: g(x) =p(x) x ? 1= x2+ x + 1 Hence, x3? 1 = (x ? 1)(x2+ x + 1). We were able to find two factors of x3? 1 because we spotted that the number 1 was a root of x3? 1. * * * * * * * x ? 1=x3? 1 * * * * * * Roots and graphs If you put a root into a polynomial, 0 comes out. That means that if ↵ is a root of p(x), then (↵,0) 2 R2is a point in the graph of p(x). These points are exactly the x-intercepts of the graph of p(x). The roots of a polynomial are exactly the x-intercepts of its graph. 135

Examples. Examples. Examples. Examples. • Below is the graph of a polynomial p(x). The graph intersects the x-axis at 2 and 4, so 2 and 4 must be roots of p(x). That means that (x?2) and (x ? 4) are factors of p(x). and (x ? 4) are factors of p(x). • Below is the graph of a polynomial p(x). The graph intersects the x-axis at 2 and 4, so 2 and 4 must be roots of p(x). That means that (x?2) Examples. • Below is the graph of a polynomial p(x). The graph intersects the x-axis at 2 and 4, so 2 and 4 must be roots of p(x). That means that (x?2) and (x ? 4) are factors of p(x). and (x ? 4) are factors of p(x). • Below is the graph of a polynomial p(x). The graph intersects the x-axis at 2 and 4, so 2 and 4 must be roots of p(x). That means that (x?2) • Below is the graph of a polynomial p(x). The graph intersects the x-axis at 2 and 4, so 2 and 4 must be roots of p(x). That means that (x?2) and (x ? 4) are factors of p(x). • Below is the graph of a polynomial q(x). The graph intersects the x-axis at ?3, 2, and 5, so ?3, 2, and 5 are roots of q(x), and (x+3), (x?2), x-axis at ?3, 2, and 5, so ?3, 2, and 5 are roots of q(x), and (x+3), (x?2), • Below is the graph of a polynomial q(x). The graph intersects the x-axis at ?3, 2, and 5, so ?3, 2, and 5 are roots of q(x), and (x+3), (x?2), and (x ? 5) are factors of q(x). and (x ? 5) are factors of q(x). and (x ? 5) are factors of q(x). • Below is the graph of a polynomial q(x). The graph intersects the • Below is the graph of a polynomial q(x). The graph intersects the x-axis at ?3, 2, and 5, so ?3, 2, and 5 are roots of q(x), and (x+3), (x?2), x-axis at ?3, 2, and 5, so ?3, 2, and 5 are roots of q(x), and (x+3), (x?2), • Below is the graph of a polynomial q(x). The graph intersects the and (x ? 5) are factors of q(x). and (x ? 5) are factors of q(x). 107 * * * * * * * 5 * * * * * * 107 5 136

Degree of a product is the sum of degrees of the factors Let’s take a look at some products of polynomials that we saw before in the chapter on “Basics of Polynomials”: The leading term of (2x2? 5x)(?7x + 4) is ?14x3. This is an example of a degree 2 and a degree 1 polynomial whose product equals 3. Notice that 2 + 1 = 3 The product 5(x?2)(x+3)(x2+3x?7) is a degree 4 polynomial because its leading term is 5x4. The degrees of 5, (x ? 2), (x + 3), and (x2+ 3x ? 7) are 0, 1, 1, and 2, respectively. Notice that 0 + 1 + 1 + 2 = 4. The degrees of (2x3? 7), (x5? 3x + 5), (x ? 1), and (5x7+ 6x ? 9) are 3, 5, 1, and 7, respectively. The degree of their product, (2x3? 7)(x5? 3x + 5)(x ? 1)(5x7+ 6x ? 9), equals 16 since its leading term is 10x16. Once again, we have that the sum of the degrees of the factors equals the degree of the product: 3+5+1+7 = 16. These three examples suggest a general pattern that always holds for fac- tored polynomials (as long as the factored polynomial does not equal 0): If a polynomial p(x) is factored into a product of polynomials, then the degree of p(x) equals the sum of the degrees of its factors. Examples. • The degree of (4x3+ 27x ? 3)(3x6? 27x3+ 15) equals 3 + 6 = 9. • The degree of ?7(x + 4)(x ? 1)(x ? 3)(x ? 3)(x2+ 1) equals 0 + 1 + 1 + 1 + 1 + 2 = 6. Degree of a polynomial bounds the number of roots Suppose p(x) is a polynomial that has n roots, and that p(x) is not the constant polynomial p(x) = 0. Let’s name the roots of p(x) as ↵1,↵2,...,↵n. Any root of p(x) gives a linear factor of p(x), so p(x) = (x ? ↵1)(x ? ↵2)···(x ? ↵n)q(x) for some polynomial q(x). Because the degree of a product is the sum of the degrees, the degree of p(x) is at least n. 137

The degree of p(x) (if p(x) 6= 0) is greater than or equal to the number of roots that p(x) has. Examples. • 5x4? 3x3+ 2x ? 17 has at most 4 roots. • 4x723? 15x52+ 37x14? 7 has at most 723 roots. • Aside from the constant polynomial p(x) = 0, if a function has a graph that has infinitely many x-intercepts, then the function cannot be a polynomial. If it were a polynomial, its number of roots (or alternatively, its number of x-intercepts) would be bounded by the degree of the polynomial, and thus there would only be finitely many x-intercepts. To illustrate, if you are familiar with the graphs of the functions sin(x) and cos(x), then you’ll recall that they each have infinitely many x-intercepts. Thus, they cannot be polynomials. (If you are unfamiliar with sin(x) and cos(x), then you can ignore this paragraph.) 138

Exercises Exercises Exercises Exercises Exercises 1.) Name two roots of the polynomial (x ? 1)(x ? 2). 2.) Name two roots of the polynomial ?(x + 7)(x ? 3). 3.) Name four roots of the polynomial ?2 It will help with #4-6 to know that each of the polynomials from those It will help with #4-6 to know that each of the polynomials from those problems has a root that equals either ?1, 0, or 1. 4.) Write x3+ 4x ? 5 as a product of a linear and a quadratic polynomial. 1.) Name two roots of the polynomial (x ? 1)(x ? 2). 2.) Name two roots of the polynomial ?(x + 7)(x ? 3). 3.) Name four roots of the polynomial ?2 1.) Name two roots of the polynomial (x ? 1)(x ? 2). 2.) Name two roots of the polynomial ?(x+7)(x?3)(x4+x3+2x2+x+1). 3.) Name four roots of the polynomial ?2 It will help with #4-6 to know that each of the polynomials from those problems has a root that equals either ?1, 0, or 1. Remember that if ↵ is a root of p(x), then 4.) Write x3+ 4x ? 5 as a product of a linear and a quadratic polynomial. 5.) Write x3+x as a product of a linear and a quadratic polynomial. (Hint: you could use the distributive law here.) you could use the distributive law here.) 6.) Write x5+ 3x4+ x3? x2? x ? 1 as a product of a linear and a quartic polynomial. polynomial. 1.) Name two roots of the polynomial (x ? 1)(x ? 2). 2.) Name two roots of the polynomial ?(x + 7)(x ? 3). 3.) Name four roots of the polynomial ?2 It will help with #4-6 to know that each of the polynomials from those problems has a root that equals either ?1, 0, or 1. problems has a root that equals either ?1, 0, or 1. 4.) Write x3+ 4x ? 5 as a product of a linear and a quadratic polynomial. 5.) Write x3+x as a product of a linear and a quadratic polynomial. (Hint: you could use the distributive law here.) you could use the distributive law here.) 1.) Name two roots of the polynomial (x ? 1)(x ? 2). 2.) Name two roots of the polynomial ?(x + 7)(x ? 3). 3.) Name four roots of the polynomial ?2 It will help with #4-6 to know that each of the polynomials from those problems has a root that equals either ?1, 0, or 1. 4.) Write x3+ 4x ? 5 as a product of a linear and a quadratic polynomial. 5.) Write x3+x as a product of a linear and a quadratic polynomial. (Hint: 5.) Write x3+x as a product of a linear and a quadratic polynomial. (Hint: 5(x +7 3)(x +1 2)(x ?4 3)(x ?9 2). 5(x +7 3)(x +1 2)(x ?4 3)(x ?9 2). 5(x+7 3)(x+1 2)(x?4 3)(x?9 2)(x2+1). 3)(x +1 5(x +7 3)(x +1 2)(x ?4 3)(x ?9 2). 2). 5(x +7 2)(x ?4 3)(x ?9 p(x) x?↵is a polynomial and p(x) = (x ? ↵)p(x) 4.) Write x3+ 4x ? 5 as a product of a linear and a quadratic polynomial. 5.) Write x3+x as a product of a linear and a quadratic polynomial. (Hint: you could use the distributive law here.) 6.) Write x5+ 3x4+ x3? x2? x ? 1 as a product of a linear and a quartic polynomial. polynomial. 7.) The graph of a polynomial p(x) is drawn below. Identify as many roots and factors of p(x) as you can. and factors of p(x) as you can. x?↵. 6.) Write x5+ 3x4+ x3? x2? x ? 1 as a product of a linear and a quartic 6.) Write x5+ 3x4+ x3? x2? x ? 1 as a product of a linear and a quartic 6.) Write x5+ 3x4+ x3? x2? x ? 1 as a product of a linear and a quartic polynomial. 7.) The graph of a polynomial p(x) is drawn below. Identify as many roots and factors of p(x) as you can. and factors of p(x) as you can. 7.) The graph of a polynomial p(x) is drawn below. Identify as many roots 7.) The graph of a polynomial p(x) is drawn below. Identify as many roots 7.) The graph of a polynomial p(x) is drawn below. Identify as many roots and factors of p(x) as you can. 8.) The graph of a polynomial q(x) is drawn below. Identify as many roots 8.) The graph of a polynomial q(x) is drawn below. Identify as many roots 8.) The graph of a polynomial q(x) is drawn below. Identify as many roots 8.) The graph of a polynomial q(x) is drawn below. Identify as many roots and factors of q(x) as you can. and factors of q(x) as you can. 8.) The graph of a polynomial q(x) is drawn below. Identify as many roots and factors of q(x) as you can. and factors of q(x) as you can. and factors of q(x) as you can. 108 6 139 108 6

For #9-13, determine the degree of the given polynomial. 9.) (x + 3)(x ? 2) 10.) (3x + 5)(4x2+ 2x ? 3) 11.) ?17(3x2+ 20x ? 4) 12.) 4(x ? 1)(x ? 1)(x ? 1)(x ? 2)(x2+ 7)(x2+ 3x ? 4) 13.) 5(x ? 3)(x2+ 1) 14.) (True/False) 7x5+ 13x4? 3x3? 7x2+ 2x ? 1 has 8 roots. For #15-17, divide the polynomials. You can use synthetic division for #17 if you’d like. x6? 2x5+ 6x4? 10x3+ 14x2? 10x + 14 x2+ 3 15.) ?2x3+ x2+ 4x ? 6 2x ? 1 16.) ?2x3+ 4x ? 6 x ? 2 17.) 140