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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) = 15x 2 7 x 8 since q(1) = 15(1) 2 7(1) 8 = 15 7 8 = 0
- ( 2
p
- 2 2 = 0, so 2
p 2 is a root of x 2 2.
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.
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. 3 x 3 x 2 + 12x 4 = (3x 1)(x 2 + 4), so 3x 1 is a factor of 3 x 3 x 2 + 12x 4. The polynomial x 2 + 4 is also a factor of 3x 3 x 2 + 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 p q((xx)) is a polynomial, and p(x) = q(x)
(^) p(x) q(x)
. That’s the
definition of q(x) being a factor of p(x).
If p q((xx)) has a remainder, then q(x) is not a factor of p(x).
Example. In the previous chapter we saw that
6 x 2 + 5x + 1 3 x + 1
= 2x + 1
Multiplying the above equation by 3x + 1 gives
6 x 2 + 5x + 1 = (3x + 1)(2x + 1)
so 3x + 1 is a factor of 6x 2 + 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
x + 2 2
x + 15 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
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) = x 3 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
x 3 1 x 1
= x 2 + x + 1
Hence, x 3 1 = (x 1)(x 2 + x + 1). We were able to find two factors of x 3 1 because we spotted that the number 1 was a root of x 3 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 R 2 is 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.
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).
- 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).
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).
- 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).
107
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).
- 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).
5
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).
- 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).
107
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).
- 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).
5
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.
- 5 x 4 3 x 3 + 2x 17 has at most 4 roots.
- 4 x 723 15 x 52 + 37x 14 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.)
Exercises
1.) Name two roots of the polynomial (x 1)(x 2).
2.) Name two roots of the polynomial (x + 7)(x 3)(x 4 + x 3 + 2x 2 + x + 1).
3.) Name four roots of the polynomial 25 (x+ 73 )(x+ 12 )(x 43 )(x 92 )(x 2 +1).
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 p x( x↵) is a polynomial and p(x) = (x ↵) p x( x↵).
4.) Write x 3 + 4x 5 as a product of a linear and a quadratic polynomial.
5.) Write x 3 +x as a product of a linear and a quadratic polynomial. (Hint: you could use the distributive law here.)
6.) Write x 5 + 3x 4 + x 3 x 2 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.
8.) The graph of a polynomial q(x) is drawn below. Identify as many roots and factors of q(x) as you can.
139
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 ^25 (x + 73 )(x + 12 )(x 43 )(x 92 )
It will help with #4-6 to know that each of the polynomials from th problems has a root that equals either 1, 0, or 1.
4.) Write x^3 + 4 x 5 as a product of a linear and a quadratic polynom
5.) Write x^3 +x as a product of a linear and a quadratic polynomial. (H you could use the distributive law here.)
6.) Write x^5 + 3 x^4 + x^3 x^2 x 1 as a product of a linear and a qua polynomial.
7.) The graph of a polynomial p(x) is drawn below. Identify as many r and factors of p(x) as you can.
8.) The graph of a polynomial q(x) is drawn below. Identify as many r and factors of q(x) as you can.
108
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 ^25 (x + 73 )(x + 12 )(x 43 )(x 92
It will help with # 4 -6 to know that each of the polynomials from t
problems has a root that equals either 1 , 0 , or 1.
4 .) Write x^3 + 4 x 5 as a product of a linear and a quadratic polynom
5 .) Write x^3 +x as a product of a linear and a quadratic polynomial. (H
you could use the distributive law here.)
6 .) Write x^5 + 3 x^4 + x^3 x^2 x 1 as a product of a linear and a qu
polynomial.
7 .) The graph of a polynomial p(x) is drawn below. Identify as many r
and factors of p(x) as you can.
8 .) The graph of a polynomial q(x) is drawn below. Identify as many r
and factors of q(x) as you can.
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 ^25 (x + 73 )(x + 12 )(x 43 )(x 92 )
It will help with #4-6 to know that each of the polynomials from th problems has a root that equals either 1, 0, or 1.
4.) Write x^3 + 4 x 5 as a product of a linear and a quadratic polynom
5.) Write x^3 +x as a product of a linear and a quadratic polynomial. (H you could use the distributive law here.)
6.) Write x^5 + 3 x^4 + x^3 x^2 x 1 as a product of a linear and a qua polynomial.
7.) The graph of a polynomial p(x) is drawn below. Identify as many ro and factors of p(x) as you can.
8.) The graph of a polynomial q(x) is drawn below. Identify as many ro and factors of q(x) as you can.
108
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 ^25 (x + 73 )(x + 12 )(x 43 )(x (^92)
It will help with # 4 -6 to know that each of the polynomials from t problems has a root that equals either 1 , 0 , or 1.
4 .) Write x^3 + 4 x 5 as a product of a linear and a quadratic polynom
5 .) Write x^3 +x as a product of a linear and a quadratic polynomial. (H you could use the distributive law here.)
6 .) Write x^5 + 3 x^4 + x^3 x^2 x 1 as a product of a linear and a qua polynomial.
7 .) The graph of a polynomial p(x) is drawn below. Identify as many r and factors of p(x) as you can.
8 .) The graph of a polynomial q(x) is drawn below. Identify as many r and factors of q(x) as you can.
6
