5.2 Polynomials and Polynomial Functions
Determine the coefficient and degree of a monomial
Monomial: An expression that is a constant, or a product of a constant and variables that are rised to whole number powers.
Coefficient: The numerical factor in a monomial (no number means it is a 1).
Degree of a Monomial: The sum of the exponents of all variables in the monomial.
[10] Identify the coefficient & degree of -0.4a3b2c -0.4 & 3 + 2 + 1 = 6
Determine the Degree of a Polynomial and Write Polynomials in Descending Order of Degree
Polynomial: A monomial or an expression that can be written as a sum/difference of monomials.
Examples: 4x, 4x + 8, 2x2 + 5xy - 8y, 3
Polynomial in one variable: A polynomial in which every variable term has the same variable.
Binomial: A polynomial containing two terms.
Trinomial: A polynomial containing three terms.
Degree of a Polynomial: The greatest degree of any of the terms in the polynomial.
Polynomials with one variable are written in descending powers of the variable.
Example: 3x2 + 4x - 4x3 - 4 - 5x5
should be written
-5x5 + 0x4 - 4x3 + 3x2 + 4x - 4
missing terms are normally not printed, but the coefficient of a missing power is 0
Examples
Indicate the degree of each polynomial, indicate if it is a monomial, binomial, trinomial, or no special name.
[14] 1.5r4 - 3r2 + 9r trinomial, degree 4
[16] --7.1k + 2.3k3 - 8k2 - 1 rewrite as 2.3k3 – 8k2 -7.1k – 1 no name, degree 3
[22] 3m2n2 + 6m4n binomial, degree 5
Add and Subtract Polynomials
The sum of two functions, f + g, is found by (f + g)(x) = f(x) + g(x).
The difference of two functions, f – g, is found by (f – g)(x) = f(x) – g(x).
It is best to add/subtract polynomials vertically.
Insure that the vertical terms have identical variables and the same powers.
If subtracting, remember to change all of the polynomial’s signs before combining the terms.
[24] (3y2 + 7y – 3) + (4y2 + 3y + 1)
+3y2 + 7y - 3
+ ( +4y2 + 3y + 1 )
+7y2 + 10y - 2
[34] (4x3 - 3x + 4) - (6x3 - 3x2 + 5)
+4x3 - 3x + 4
- ( +6x3 - 3x2 + 5 )
+ - + - .
-2x3 + 3x2 - 3x - 1
If more than two polynomials are used in addition and/or subtractions, compute the first two and use the result with the third (suggestion).
[50] (3k3–5k2–k–1) – (2k3–3k2–k–7) + (4k3+ k2+4k+7)
3k3 – 5k2 - k - 1
- ( 2k3 – 3k2 - k - 7 )
k3 - 2k2 + 6
+ ( 4k3 + 4k2 + k + 7 )
5k3 + 2k2 + k + 13
The following portions are not testable (yet)
Classify and Graph Polynomial Functions
Polynomial Function: A function of the form f(x) = axm + bxn + ··· with a finite number of terms, where each coefficient is a real number and each exponent is a whole number.
Examples:
f(x) = 4x4 - 3x2 + 5x - 3
g(x) = 5.8x + 3.2
Constant Function: A function of the form f(x) = c where c is a real number.
The graph of a constant function is a horizontal line through (0, c).
Example:
f(x) = -2
Linear Function: A function of the form f(x) = mx + b where m and b are real numbers.
Graphs of linear functions are lines with slope m and y-intercept (0, b).
Example
f(x) = 2x – 1
m = 2
y-int = (0,-1)
Quadratic Function: A function of the form f(x) = ax2 + bx + c where a, b, and c are real numbers and a ≠ 0.
Graphs of quadratic functions are parabolas with y-intercept at (0, c).
Examples:
f(x) = 2x2 + 3x – 5 ______
g(x) = -3x2 + 2x + 6 ------
Cubic Function: A function of the form f(x) = ax3 + bx2 + cx + d where a, b, c, and d are real numbers and a ≠ 0.
Graphs of cubic functions resemble an S-shape with y-intercept at (0, d).
Examples:
f(x) = 2x3 - 5x2 + 3x + 5
g(x) = -x3 - 2x2 + 4x + 5