FUNCTIONS CAN BE CATEGORIZED, and the simplest type is a polynomial function. We will define it below. We begin with vocabulary.
1. When numbers are added or subtracted, they are called terms. This --
4x2 + 7x − 8
-- is a sum of three terms. (In algebra we speak of a "sum," even though a term may be subtracted.)
When numbers are multiplied, they are called factors. This --
1. (x + 1)(x + 2)(x + 3)
-- is a product of three factors.
2. A variable is a symbol that takes on different values. A value is a number.
Thus if x is a variable and we give it the value 4, then 5x + 1 has the value 21.
3. A constant is a symbol that has a single value. The symbols 2, 5, , are constants. When we write
y = ax2 + bx + c,
then a, b, c are arbitrary constants to which we may assign a definite value; for example,
y = 5x2 − 2x + 1.
We typically use the beginning letters of the alphabet to denote such constants. We use the letters x, y, z to denote variables.
4. A monomial in x is a single term of the form axn, where a is a
real number and n is a whole number.
The following are monomials in x:
5x3 −6.3x 2
We may say that the number 2 is a monomial in x because 2 = 2x0 = 2· 1. (Lesson 21 of Algebra.)
5. A polynomial in x is a sum of monomials in x.
Example 1. 5x3 − 4x2 + 7x − 8.
The variable, in this case x, is also called the argument of the polynomial. Here is a polynomial with argument t :
t 2 −5t + 1.
When we write a polynomial, the style is to begin with the highest exponent and go to the lowest. 4, 3, 2, 1.
(For the general form of a polynomial, see Problem 6 below.)
6. The degree of a term is the sum of the exponents of all the variables in that term.
In functions of a single variable, such as x, the degree of a term is simply the exponent.
Example 2. The term 5x3 is of degree 3 in the variable x.
Example 3. This term 2xy2z3 is of degree 1 + 2 + 3 = 6 in the variables x, y, and z.
Example 4. Here are all possible terms of the 4th degree in the variables x and y:
x4, x3y, x2y2, xy3, y4.
In each term, the sum of the exponents is 4. As the exponent of x decreases, the exponent of y increases.
Problem 1. Write all possible terms of the 5th degree in the variables x and y.
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
x5, x4y, x3y2, x2y3, xy4, y5.
7. The leading term of a polynomial is the term of highest degree.
Example 5. The leading term of this polynomial 5x3 − 4x2 + 7x − 8 is 5x3.
8. The leading coefficient of a polynomial is the coefficient of the leading term.
Example 6. The leading coefficient of that polynomial is 5.
9. The degree of a polynomial is the degree of the leading term.
Example 7. The degree of this polynomial 5x3 − 4x2 + 7x − 8 is 3.
Here is a polynomial of the first degree: x − 2.
1 is the highest exponent.
10. The constant term of a polynomial is the term of degree 0; it is the term in which the variable does not appear.
Example 8. The constant term of this polynomial 5x3 − 4x2 + 7x − 8 is −8.
The constant term of this polynomial --
ax3 + bx2 + cx + d
-- is d.
Problem 2. Which of the following is a polynomial? If an expression is a polynomial, name its degree, and say the variable that the polynomial is in.
a) x3 − 2x2 − 3x − 4 Polynomial of the 3rd degree in x.
b) 3y2 + 2y + 1 Polynomial of the 2nd degree in y.
c) x3 + 2 + 1
This is not a polynomial, because is not a
whole number power. It is x½.
d) z + 2 Polynomial of the first degree in z.
e) x2 − 2x +
Not a polynomial, because = x−1, which is not a whole number power.
Problem 3. Name the degree, the leading coefficient, and the constant term.
a) f(x) = 6x3 + 7x2 − 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) = 4x5
5th degree. Leading coefficient, 4. Constant term, 0.
d) f(h) = h2 − 7h − 5
2nd degree. Leading coefficient, 1. Constant term, −5.
Example 9. Name the degree, the leading coefficient, and the constant term:
(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· 23 = 15· 8 = 120.
And the constant term would be the product of all the constant terms: 1· (−1)· 53 = −1· 125 = −125.
Problem 4. Name the degree, the leading coefficient, and the constant term.
a) f(x) = (x − 1)(x2 + 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: 12· 13· 24 = 16.
Constant term: 22· (-3)3· 14 = 4· (−27) = −108
c) f(x) = (2x + 1)5
Degree: 5. Leading coefficient: 25 = 32. Constant term: 15 = 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:
ax3 + bx2 + 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 sub-script notation. We use one letter, such as a, and indicate different constants by means of sub-scripts. Thus, a1 ("a sub-1") will be one constant. a2 ("a sub-2") will be another. And so on. Here, then, is the general form of a polynomial of the 50th degree:
a50x50 + a49x49 + . . . + a2x2 + a1x + a0
The constant ak -- for each sub-script k (k = 0, 1, 2, . . . , 50) -- is the coefficient of xk.
Notice that there are 51 constants. The constant term a0 is the 51st.
a) Using subscript notation, write the general form of a polynomial of
a) the fifth degree in x.
a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
b) In that general form, how many constants are there? 6
c) Name the six constants of this fifth degree polynomial: x5 + 6x2 − x.
a5 = 1. a4 = 0. a3 = 0. a2 = 6. a1 = −1. a0 = 0.
a) Indicate the general form of a polynomial in x of degree n.
anxn + an−1xn−1 + . . . + a1x +a0
n is a whole number, the a's are real numbers, and an0.
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 = x2 + 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 = x2 , and y = x3.
Problem 7. Let f(x) be the function with the given, restricted domain. Describe its range.
a) f(x) = x2, −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) = x3, −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) = x4, −2 ≤ x ≤ 1
0 ≤ y ≤ 16. y goes from a low of 0, at x = 0, to a high of 16, at x = −2. x4 is very much like x2. The exponent is even.
d) f(x) = x5, −2 ≤ x ≤ 1
−32 ≤ y ≤ 1. y goes from a low of −32, at x = −2, to a high of 1, at x = 1. x5 is very much like x3. 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
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