1.   When numbers are added or subtracted, they are called terms.  This --

4x² + 7x − 8

1.    -- is a sum of three terms.

When numbers are multiplied, they are called factors.  This --

1.    (x + 1)(x + 2)(x + 3)

1.    -- is a product of three factors.

2.   A variable is a symbol that takes on values.  A value is a number.

Thus if x is the variable and has the value 4, then 5x + 1 has the value 21.

3.   A constant is a symbol that has a single value.

Example 1.  The symbols '5' and '' 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.

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, the degree of a term is simply the exponent.

Example 3.  The term  5x³  is of degree 3 in the variable x.

Example 4.  This term  2xy²z³ is of degree 1 + 2 + 3 = 6 in the variables x, y, and z. 

Example 5.  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.

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x⁵,  x⁴y,  x³y²,   x²y³,  xy⁴,  y⁵.

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

Example 6.  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 7.  The leading coefficient of that polynomial  is 5.

9.   The degree of a polynomial is the degree of the leading term.

Examples 8.  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 9.  The constant term of this polynomial  5x³ − 4x² + 7x − 8  is −8.

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 x^½.

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

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 10.   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.

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 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 a₀ is the 51st.

Problem 5.

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.

Problem 6.

a)  Indicate the general form of a polynomial in x of degree n.

a_nxⁿ + a_n−1xⁿ⁻¹ + . . . + a₁x +a₀

n is a whole number, and a_n0.

c)  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