6

POLYNOMIAL FUNCTIONS

Definition of a polynomial in *x*

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 --

4*x*^{2} + 7*x* − 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 5*x* + 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* = *a x*

then *a, b, c* are *arbitrary constants* to which we may assign a definite value; for example,

*y* = 5*x*^{2} − 2*x* + 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 *ax*^{n}, where *a* is a

real number and *n* is a whole number.

The following are monomials in *x*:

5*x*^{3} −6**.**3*x* 2

We may say that the number 2 is a monomial in *x* because 2 = 2*x*^{0} = 2**·** 1. (Lesson 21 of Algebra.)

5. A polynomial in *x* is a sum of monomials in *x*.

**Example 1.** 5*x*^{3} − 4*x*^{2} + 7*x* − 8.

The variable, in this case *x*, is also called the argument of the polynomial. Here is a polynomial with argument *t* :

*t* ^{2} −5*t* + 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 5*x*^{3} is of
degree 3 in the variable *x*.

**Example 3.** This term 2*x**y*^{2}*z*^{3} 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*:

*x*^{4}, *x*^{3}*y*, *x*^{2}*y*^{2}, *x**y*^{3}, *y*^{4}.

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").

*x*^{5},
*x*^{4}*y*, *x*^{3}*y*^{2}, *x*^{2}*y*^{3}, *x**y*^{4}, *y*^{5}.

7. The leading term of a polynomial is the term of highest degree.

**Example 5.** The leading term of this polynomial 5*x*^{3} − 4*x*^{2} + 7*x* − 8 is 5*x*^{3}.

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 5*x*^{3} − 4*x*^{2} + 7*x* − 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 5*x*^{3} − 4*x*^{2} + 7*x* − 8
is −8.

The constant term of this polynomial --

*a**x*^{3} + *bx*^{2} + *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) *x*^{3} − 2*x*^{2} − 3*x* − 4
Polynomial of the 3rd degree in *x*.

b) 3*y*^{2} + 2*y* + 1
Polynomial of the 2nd degree in *y*.

c) *x*^{3} + 2 + 1
This is not a polynomial, because is not a

whole number power. It is *x*^{½}.

d) *z* + 2 Polynomial of the first degree in *z*.

e) *x*^{2} − 2*x* +

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*) = 6*x*^{3} + 7*x*^{2} − 3*x* + 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*) = 4*x*^{5}

5th degree. Leading coefficient, 4. Constant term, 0.

d) *f*(*h*) = *h*^{2} − 7*h* − 5

2nd degree. Leading coefficient, 1. Constant term, −5.

**Example 9.** Name the degree, the leading coefficient, and the constant term:

(5*x* + 1)(3*x* − 1)(2*x* + 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,

(5*x* + 1)(3*x* − 1)(2*x* + 5)^{3} = (5*x* + 1)(3*x* − 1)(2*x* + 5)(2*x* + 5)(2*x* + 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}(2*x* + 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*) = (2*x* + 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:

*a**x*^{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 sub-script notation. We use one letter, such as *a*, and indicate different constants by means of sub-scripts. Thus, *a*_{1} ("*a* sub-1") will be one constant. *a*_{2} ("*a* sub-2") 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 sub-script *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} + 6*x*^{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* = 2*x* + 1, is called a linear function; while a
polynomial function of the second degree, such as *y* = *x*^{2} + 3*x* − 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.

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

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