Download MTH 251 Differential Calculus

MTH 070
Elementary Algebra
Chapter 5 – Exponents, Polynomials
and Applications
Section 5.3 – Introduction to
Polynomials and
Polynomial Functions
Copyright © 2010 by Ron Wallace, all rights reserved.
Vocabulary

Term



A number: 17
A variable: x
A product: -2x3



Coefficient of a Term

The “largest” constant factor of a term.



Positive exponents only!
No addition, subtraction, or division!
That is, the number part of the term
What about x, -x, x5, etc.?
Degree of a Term

The number of variable factors of the term.
Vocabulary

Polynomial

A term or sum of terms.



Term of a polynomial with highest degree.
Leading Coefficient


Convention: Put terms in order by degree.
Leading Term


Note: Subtraction is considered adding the opposite.
Coefficient of the leading term.
Degree of the Polynomial

Degree of the leading term.
Vocabulary

Monomial



Binomial



A polynomial with one term.
5x3
A polynomial with two terms.
3x2 – 5
Trinomial


A polynomial with three terms.
x2 – 4x + 3
Polynomial Expressions
2 x  x  5x  8x  3
4

3
2
For the above polynomial, determine …







The number of terms:
The degree of the second term:
The degree of the polynomial:
The leading coefficient:
The coefficient of the second term:
The coefficient of the linear term:
Is this polynomial a monomial, binomial or
trinomial?
Evaluating a Polynomial

Given a polynomial and a value for its
variable … substitute the value for the
variable and do the arithmetic.

Example 1:


Determine the value of x2 – 4x + 3 when x = 2
Example 2:

Determine the value of x2 – 4x + 3 when x = –2
A Little Trick for Evaluating Polynomials
Determine the value of x2 – 4x + 3 when x = –2
2
1
4
3
A Little Trick for Evaluating Polynomials
Determine the value of x2 – 4x + 3 when x = –2
2
1
1
4
3
A Little Trick for Evaluating Polynomials
Determine the value of x2 – 4x + 3 when x = –2
2
1
4
2
1
3
A Little Trick for Evaluating Polynomials
Determine the value of x2 – 4x + 3 when x = –2
2
1
4
1
2
6
3
A Little Trick for Evaluating Polynomials
Determine the value of x2 – 4x + 3 when x = –2
2
1
4
3
12
1
2
6
A Little Trick for Evaluating Polynomials
Determine the value of x2 – 4x + 3 when x = –2
2
1
4
3
1
2
6
12
15
A Little Trick for Evaluating Polynomials
Determine the value of 5x3 - 7x2 - 2 when x = 3
Functions … a review from 3.6

Function … a named expression that gives
only one result for each value of the
variable.

Notation: f(x) = an-expression-using-x




Read as “f of x equals …”
Doesn’t have to be f … g(x); h(x); p(x) …
Doesn’t have to be x … f(a); g(m); d(t) …
Evaluating a Function

f(3) means replace the variable in the expression
with 3 and do the arithmetic.
Polynomial Functions
A polynomial function is a function where
the expression is a polynomial.
 Example:



Linear Function


Polynomial function of degree 1
Quadratic Function


P(x) = 2x2 – 4x + 3
Polynomial function of degree 2
Cubic Function

Polynomial function of degree 3
Polynomial Functions
If P(x) = 2x2 + 4x + 3,
find P(0) & P(1) & P(–5)
Adding Polynomials

“Combine Like Terms”


i.e. Terms with the same variables can be
combined by adding their coefficients.
Order of terms in the answer?



Descending order by degree
Ascending order by degree
Match the problem!
Adding Polynomials

Example …
 2x
4
 5 x  3   x  7 x  4 x  6 
2
3
2
Review: Subtracting Signed Numbers

“Add the Opposite”


a – b = a + (–b)
Opposite?



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The number the same distance from zero on
the other side of zero.
–(5) = –5
–(–5) = 5
Essentially, multiplication by –1
Opposite of a Polynomial

If p(x) is a polynomial, then its opposite is …
 –p(x) = (-1)p(x)

Example:
 –(3x – 4) = (–1)(3x – 4) = –3x + 4

That is: Change the sign of every term.
Subtracting Polynomials

If p(x) and q(x) are polynomials, then

p(x) – q(x) = p(x) + (–q(x))

i.e. Add the opposite of the polynomial
that follows the subtraction sign!
Subtracting Polynomials

Example …
 2x
4
 5 x  3   x  7 x  4 x  6 
2
3
2