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Chapter 29
Polynomials
c H. Feiner 2011
29.1
Basics
A polynomial is a particular expression. It is a sum or difference of special terms. Each term is the product
of a coefficient (a numerical factor) and one or more variables raised to a whole number.
2x3 is such a term. 2 is the coefficient and the exponent 3 is the whole number exponent. The exponent is
called the degree.
−2x5 y4 is another such term. The coefficient is −2 and the degree 9 (the sum of the exponents).
We shall concentrate on polynomials in one variable. The exponents are written in descending (or increasing)
order of exponents.
5x7 − 6x4 + x3 + 9x − 1 is fine. So is −1 + 9x + x3 − 6x4 + 5x7 , but not 5x7 + 9x + x3 − 6x4 − 1.
The polynomial 5x6 − 4x3 + 7x2 + 8 has four terms. The leading coefficient (the coefficient of the largest
degree) is 5. The degree of the polynomial (the largest exponent) is 6.
The coefficient of the third degree term is −4 (this is also the coefficient of the second term.)
The coefficient of the second degree term is 7. The coefficient of the sixth degree term is 0.
The constant is 8.
The general form of a sixth degree polynomial is
a6 x6 + a5 x5 + a4 x4 + a3 x3 + a22 + a1 x + a0
Matching our last expression above with 5x6 + 0x5 + 0x4 (−4)x3 + 7x2 + 0x + 8 leads to a6 = 5, a5 = 0,
1
2
CHAPTER 29. POLYNOMIALS
a4 = 0, a3 = −4, a2 = 7, a1 = 0, a0 = 8.
6x4 + 5x−3 − 2x + 9 is not a polynomial because the exponent −3 is not a whole number.
4
4
6x + 5
− 2x + 9 is not a polynomial because the exponent −3 is not a whole number. Recall that
x3
1
= x−3 .
x3
2
6x4 + 5−2 x−3 − 7 x + 9 is a polynomial because the exponents are all whole numbers and there is no
3
restriction on exponents.
A polynomial with three terms is a trinomial. (A tricycle has three wheels).
A polynomial with two terms is a binomial. (A bicycle has two wheels).
A polynomial with one term is a monomial. (A monorail has one rail.)
4x2 y3 − 5xy7 − 8x + 9y10 − 1 is another example of a polynomial.
Polynomials are used in many mathematical an scientific situations. The speed ball you favorite baseball
player throws follows a trajectory like f(x) = 9.8x2 + 49x + 5.
Evaluating a polynomial means finding the value of the polynomial when x is given.
29.2
Examples
Example 1:
Give an example of a fifth degree binomial with leading coefficient 3. (The answer is not unique.)
Solution:
3x5 + axm is such a polynomial. a can be any real number and m is a whole number 6= 5.
Example 2:
Evaluate the fourth degree polynomial f(x) = −2x4 + 5x2 − 3 if x = −2.
Solution:
f(x)
f(−2)
=
=
=
=
=
−2x4 + 5x2 − 3
−2(−2)4 + 5(−2)2 − 3
−2(16) + 5(4) − 3
−32 + 20 − 3
−15
Example 3:
A projectile is thrown upward with velocity 48
ft
from a platform 32 ft above ground. The equation of its
sec
3
29.3. EXERCISE 29
path is H(t) = −16t2 + 48t + 32 ft.
How high above ground will the projectile be in 3 seconds after launch?
Solution:
H(t) =
H(3) =
=
=
=
−16t2 + 48t + 32
−16(3)2 + 48(3) + 32
−16(9) + 144 + 32
−144 + 144 + 32
32
The projectile will be 32 ft above ground.
29.3
Exercise 29
1. Give an example of a sixth degree trinomial with leading coefficient 7. (The answer is not unique.)
2. Evaluate the sixth degree polynomial f(x) = −2x6 + 5x3 − 5 if x = −1.
3. A ball is thrown upward with velocity 100
its path is H(t) = −16t2 + 10t + 200 ft.
ft
from a platform 200 ft above ground. The equation of
sec
How high above ground will the projectile be in 4 seconds after launch?
STOP!
1. Give an example of a sixth degree trinomial with leading coefficient 7. (The answer is not unique.)
Solution:
7x6 + axm + bxn is such a polynomial. a can be any real number and m, n are a whole number 6= 6
and m 6= n.
2. Evaluate the sixth degree polynomial f(x) = −2x6 + 5x3 − 5 if x = −1.
Solution:
f(x) =
f(−1) =
=
=
−2x6 + 5x3 − 5
−2(−1)6 + 5(−1)3 − 5
−2(1) + 5(−1) − 5
−17
Example 3:
A ball is thrown upward with velocity 100
its path is H(t) = −16t2 + 10t + 200 ft.
ft
from a platform 200 ft above ground. The equation of
sec
How high above ground will the projectile be in 4 seconds after launch?
Solution:
4
CHAPTER 29. POLYNOMIALS
H(t) =
H(4) =
=
=
=
=
−16t2 + 10t + 200
−16(4)2 + 10(3) + 200
−16(16) + 30 + 200
−144 + 30 + 200
−114 + 200
86
The projectile will be 86 ft above ground.