6 THE VOCABULARY OF

e) x^{2} − 2x +  1 x 
Not a polynomial, because  1 x 
= x^{−1}, which is not a whole 
number power.
Problem 3. Name the degree, the leading coefficient, and the constant term.
a) f(x) = 6x^{3} + 7x^{2} − 3x + 1
3rd degree. Leading coefficient, 6. Constant term, 1.
b) g(x) = −x + 2
1st degree. Leading coefficient, −1. Constant term, 2.
c) h(x) = 4x^{5}
5th degree. Leading coefficient, 4. Constant term, 0.
d) f(h) = h^{2} − 7h − 5
2nd degree. Leading coefficient, 1. Constant term, −5.
Example 9. Name the degree, the leading coefficient, and the constant term of (5x + 1)(3x − 1)(2x + 5)^{3}.
If we were to multiply out, then the degree of the product would be the sum of the degrees of each factor: 1 + 1 + 3 = 5. For,
(5x + 1)(3x − 1)(2x + 5)^{3} = (5x + 1)(3x − 1)(2x + 5)(2x + 5)(2x + 5).
The leading coefficient would be the product of all the leading coefficients: 5· 3· 2^{3} = 15· 8 = 120.
And the constant term would be the product of all the constant terms: 1· (−1)· 5^{3} = −1· 125 = −125.
Problem 4. Name the degree, the leading coefficient, and the constant term.
a) f(x) = (x − 1)(x^{2} + x − 6)
Degree: 3. Leading coefficient: 1. Constant term: 6.
b) g(x) = (x + 2)^{2}(x − 3)^{3}(2x + 1)^{4}
Degree: 9. Leading coefficient: 1^{2}· 1^{3}· 2^{4} = 16.
Constant term: 2^{2}· (3)^{3}· 1^{4} = 4· (−27) = −108
c) f(x) = (2x + 1)^{5}
Degree: 5. Leading coefficient: 2^{5} = 32. Constant term: 1^{5} = 1.
d) h(x) = x(x − 2)^{5}(x + 3)^{2}
Degree: 8. Leading coefficient: 1. Constant term: 0.
11. The general form of a polynomial shows the terms of all possible degree. Here, for example, is the general form of a polynomial of the third degree:
ax^{3} + bx^{2} + cx + d
Notice that there are four constants: a, b, c, d.
In the general form, the number of constants, because of the term of degree 0, is always one more than the degree of the polynomial.
Now, to indicate a polynomial of the 50th degree, we cannot indicate the constants by resorting to different letters. Instead, we use subscript notation. We use one letter, such as a, and indicate different constants by means of subscripts. Thus, a_{1} ("a sub1") will be one constant. a_{2} ("a sub2") will be another. And so on. Here, then, is the general form of a polynomial of the 50th degree:
a_{50}x^{50} + a_{49}x^{49} + . . . + a_{2}x^{2} + a_{1}x + a_{0}
The constant a_{k}  for each subscript k (k = 0, 1, 2, . . . , 50)  is the coefficient of x^{k}.
Notice that there are 51 constants. The constant term a_{0} is the 51st.
Problem 5.
a) Using subscript notation, write the general form of a polynomial of
a) the fifth degree in x.
a_{5}x^{5} + a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}
b) In that general form, how many constants are there? 6
c) Name the six constants of this fifth degree polynomial: x^{5} + 6x^{2} − x.
a_{5} = 1. a_{4} = 0. a_{3} = 0. a_{2} = 6. a_{1} = −1. a_{0} = 0.
Problem 6.
a) Indicate the general form of a polynomial in x of degree n.
a_{n}x^{n} + a_{n−1}x^{n−1} + . . . + a_{1}x +a_{0}
n is a whole number, the a's are real numbers, and a_{n}0.
b) A polynomial of degree n has how many constants? n + 1
12. A polynomial function has the form
y = A polynomial
A polynomial function of the first degree, such as y = 2x + 1, is called a linear function; while a polynomial function of the second degree, such as y = x^{2} + 3x − 2, is called a quadratic.
Domain and range
The natural domain of any polynomial function is
− < x < .
x may take on any real value. Consider the graphs of y = x^{2} , and y = x^{3}.
Problem 7. Let f(x) be the function with the given, restricted domain. Describe its range.
(If you are not viewing this page with Internet Explorer 6, then your browser may not be able to display the symbol ≤, "is less than or equal to;" or ≥, "is greater than or equal to.")
a) f(x) = x^{2}, −3 ≤ x ≤ 3
0 ≤ y ≤ 9. y goes from a low of 0 (at x = 0) to a high of 9 (at both −3 and 3).
b) f(x) = x^{3}, −3 ≤ x ≤ 3
−27 ≤ y ≤ 27. y goes from a low of −27 (at x = −3) to a high of 27 (at x = 3).
c) f(x) = x^{4}, −2 ≤ x ≤ 1
0 ≤ y ≤ 16. y goes from a low of 0, at x = 0, to a high of 16, at x = −2. x^{4} is very much like x^{2}. The exponent is even.
d) f(x) = x^{5}, −2 ≤ x ≤ 1
−32 ≤ y ≤ 1. y goes from a low of −32, at x = −2, to a high of 1, at x = 1. x^{5} is very much like x^{3}. The exponent is odd.
In the following Topics we will focus on the graphs of these polynomial functions.
Next Topic: The roots, or zeros, of a polynomial
Please make a donation to keep TheMathPage online.
Even $1 will help.
Copyright © 2016 Lawrence Spector
Questions or comments?
Email: themathpage@nyc.rr.com