THE VOCABULARY OF
POLYNOMIAL FUNCTIONS
From: http://www.themathpage.com/aPreCalc/polynomial.htm#poly1
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Terms and factors
Variables versus constants
Definition of a polynomial in x
Degree of a term
General form of a polynomial
Domain and range
FUNCTIONS CAN BE CATEGORIZED, and the simplest type is a polynomial function. We will define it below.
We will begin with vocabulary.
1. When numbers are added or subtracted, they are called terms. This -4x² + 7x − 8
-- is a sum of three terms.
When numbers are multiplied, they are called factors. This -(x + 1)(x + 2)(x + 3)
-- is a product of three factors.
2. A variable is a symbol that takes on values. A value is a number.
3. A constant is a symbol that has a single value.
Example. The symbols '5' and 'π' are constants.
The beginning letters of the alphabet a, b, c, etc. are typically used to denote constants, while the letters x,
y, z , are typically used to denote variables. For example, if we write
y = ax² + bx + c,
we mean that a, b, c are constants (i.e. fixed numbers), and that x and y are variables.
4. A polynomial in x is a sum of whole number powers of the variable x.
Example. 5x³ − 4x² + 7x − 8
The exponents here are the whole numbers 3, 2, 1, and 0. For, 8 = 8x0, i.e. 8· 1.
The whole numbers, recall, are the non-negative integers: 0, 1, 2, 3, 4, etc.
In addition, the coefficients in a polynomial must be real.
The variable that the polynomial is in, is also called the argument of the polynomial. Here is a polynomial in
argument t :
t ² −5t + 1
5. The degree of a term is the sum of the exponents of all the variables in that term.
In functions of a single variable, the degree of a term is simply the exponent.
Example 1. The term 5x³ is of degree 3 in the variable x.
Example 2. The term 2xy²z³ is of degree 1 + 2 + 3 = 6 in the variables x, y, and z.
Example 3. 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.
6. The leading term of a polynomial is the term of highest degree.
Example. The leading term of this polynomial 5x³ − 4x² + 7x − 8 is 5x³.
7. The leading coefficient of a polynomial is the coefficient of the leading term.
Example. The leading coefficient of that polynomial is 5.
8. The degree of a polynomial is the degree of the leading term.
Examples. The degree of this polynomial 5x³ − 4x² + 7x − 8 is 3.
x − 2 is a polynomial of the first degree. 1 is the highest exponent.
9. The constant term of a polynomial is the term of degree 0; it is the term in which the variable does not
appear.
Example. The constant term of this polynomial 5x³ − 4x² + 7x − 8 is −8.
10. 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 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.
Example. 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· 2³ = 15· 8 = 120.
And the constant term would be the product of all the constant terms: 1· (−1)· 5³ = −1· 125 = −125.
11. 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.
Test your Reading:
Name _____________________________________
Problem 1. Write all possible terms of the 5th degree in the variables x and y.
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
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b) 3y² + 2y + 1
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c) x3 + 2 x + 1
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d) z + 2
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e) x² − 2x +
1
x
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Problem 3.
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Name the degree, the leading coefficient, and the constant term.
a) f(x) = 6x3 + 7x² − 3x + 1 ______________________________________________
b) g(x) = −x + 2 ______________________________________________
c) h(x) = 4x5 ______________________________________________
d) f(h) = h² − 7h − 5 ______________________________________________
Problem 4. Name the degree, the leading coefficient, and the constant term.
a) f(x) = (x − 1)(x² + x − 6) ______________________________________________
b) g(x) = (x + 2)²(x − 3)3(2x + 1)4 ______________________________________________
c) f(x) = (2x + 1)5 ______________________________________________
d) h(x) = x(x − 2)5(x + 3)² ______________________________________________
Problem 5.
a) Using subscript notation, write the general form of a polynomial of the fifth degree in x.
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b) In that general form, how many constants are there?
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c) Name the six constants of this fifth degree polynomial: x5 + 6x² − x.
a5 = ___
a4 = ___
a3 = ___
a2 = ___
a1 = ___
a0 = ___
Problem 6.
a) Indicate the general form of a polynomial in x of degree n.
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b) A polynomial of degree n has how many constants?
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Problem 7. Let f(x) be the function with the given, restricted domain. Describe its range.
a) f(x) = x², −3 ≤ x ≤ 3
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b) f(x) = x3,
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−3 ≤ x ≤ 3
c) f(x) = x4, −2 ≤ x ≤ 1
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d) f(x) = x5, −2 ≤ x ≤ 1
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Answers:
1.
x5, x4y, x3y2,
x2y3, xy4, y5.
2.
a) Polynomial of the 3rd degree in x.
b) 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) Not a polynomial, because
3.
1
x
= x−1, which is not a whole number power.
a) 3rd degree. Leading coefficient, 6. Constant term, 1.
b) 1st degree. Leading coefficient, −1. Constant term, 2.
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c) 5th degree. Leading coefficient, 4. Constant term, 0.
d) 2nd degree. Leading coefficient, 1. Constant term, −5.
4. a) Degree: 3. Leading coefficient: 1. Constant term: 6.
b) Degree: 9. Leading coefficient: 1²· 1³· 24 = 16. Constant term: 2²· (-3)³· 14 = 4· (−27) = −108
c) Degree: 5. Leading coefficient: 25 = 32. Constant term: 15 = 1.
d) Degree: 8. Leading coefficient: 1. Constant term: 0.
5.
a) a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
b) 6
c) a5 = 1. a4 = 0. a3 = 0. a2 = 6. a1 = −1. a0 = 0.
6.
a) anxn + an−1xn−1 + . . . + a1x +a0
n is a whole number, and an  0.
b) n + 1
7. a) 0 ≤ y ≤ 9. y goes from a low of 0 (at x = 0) to a high of 9 (at both −3 and 3).
b) −27 ≤ y ≤ 27. y goes from a low of −27 (at x = −3) to a high of 27 (at x = 3).
c) 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) −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.