Transcript Slide 1

5.4 Unit – 3.3 - Polynomials
2
1
Identify terms and coefficients.
2
Know the vocabulary for polynomials.
3
Add like terms.
4
Add and subtract polynomials.
5
Evaluate polynomials.
Identify terms and coefficients.
In an expression such as
4x3  6x2  5x  8,
the quantities 4x3, 6x2, 5x, and 8 are called terms. In the first term
4x3, the number 4 is called the coefficient, of x3. In the same way, 6
is the coefficient of x2 in the term 6x2, and 5 is the coefficient of x in
the term 5x. The constant term is 8 .
Slide 5.4-4
EXAMPLE 1 Identifying Coefficients
Name the coefficient of each term in the expression
2 x  x.
3
Solution:
2, 1
Slide 5.4-5
Know the vocabulary for polynomials.
A polynomial in x is a term or the sum of a finite number of terms of
the form axn, for any real number a and any whole number (no
negative, no fraction) n. For example,
16 x8  7 x6  5x4  3x2  4
Polynomial in x
is a polynomial in x. (The 4 can be written as 4x0.) This polynomial is
written in standard form, since the exponents on x decrease from left
to right.
By contrast,
2x 3  x 2 
1
x
Not a Polynomial
is not a polynomial in x, since x appears in a denominator.
A polynomial can be defined using any variable and not just x. In fact,
polynomials may have terms with more than one variable.
Slide 5.4-10
Know the vocabulary for polynomials. (cont’d)
The degree of a term is the sum of the exponents on the variables.
For example 3x4 has degree 4, while the term 5x (or 5x1) has degree 1,
−7 has degree 0 ( since −7 can be written −7x0).
The degree of a polynomial is the greatest degree term of the
polynomial. For example 3x4 + 5x2 + 6 is of degree 4.
Slide 5.4-11
Know the vocabulary for polynomials. (cont’d)
Three types of polynomials are common and given special names. A
polynomial with only one term is called a monomial. (Mono- means
“one,” as in monorail.) Examples are
9m,
6y5 ,
a2 ,
and
6.
Monomials
A polynomial with exactly two terms is called a binomial. (Bi- means
“two,” as in bicycle.) Examples are
9 x4  9 x3 , 8m2  6m,
and
3m5  9m2 .
Binomials
A polynomial with exactly three terms is called a trinomial. (Tri- means
“three,” as in triangle.) Examples are
19 2 8
y  y  5,
9m  4m  6,
3
3
3
2
and
3m5  9m2  2.
Trinomials
Slide 5.4-12
Know the vocabulary for polynomials. (cont’d)
Degree
Name
Example
0
Constant
5
1
Linear
2xy + 4
2
Quadratic
4x2 – 7x + 2
3
Cubic
x3 – 2x2 – 2x + 1
4
Quartic
-8x4 – 7x2 + 3x - 4
5
Quintic
3x5 – 5x4 + 2x3 – 4x2 +10y
Number of Terms
Name
Example
1
Monomial
3x2
2
Binomial
5x + 4x3
3
Trinomial
3x + 4x3 - 7
More than 3
Polynomial
5x5 + 4x4 - 2x3 + 8x2 - x - 1
EXAMPLE 3 Classifying Polynomials
Write polynomial in standard form, give the degree, and tell whether
the polynomial is a monomial, binomial, trinomial.
3x + 5x3 - 4
Solution:
5x3 + 3x – 4
Degree 3 or cubic,
trinomial
Slide 5.4-13
Add like terms.
like terms have exactly the same combinations of variables, with the
same exponents on the variables. Only the coefficients may differ.
19m3 and 14m3
6 y9 ,  37 y9 , and y9
3pq and  2 pq
2xy
2
and  xy
Examples of
like terms
2
We combine, or add, like terms by adding their coefficients.
Slide 5.4-7
EXAMPLE 2 Adding Like Terms
Simplify by adding like terms.
r 2  3r  5r 2
Solution:
6r 2  3r
Unlike terms cannot be combined. Unlike terms have different variables or
different exponents on the same variables.
Slide 5.4-8
Add and subtract polynomials.
Polynomials may be added, subtracted, multiplied, and divided.
Adding Polynomials
To add two polynomials, add like terms.
Subtracting Polynomials
To subtract two polynomials, change all the signs in the second
polynomial and add the result to the first polynomial.
Slide 5.4-17
EXAMPLE 5 Adding Polynomials Vertically
Add.
4 x  3x  2 x
and
x2  2 x  5
and
3
2
6 x3  2 x 2  3x
4 x2  2
Solution:
4 x3  3x 2  2 x
3
2
+ 6 x  2 x  3x
10x3  x2  x
x2  2 x  5
2
2
+ 4x
5x2  2 x  3
Slide 5.4-18
EXAMPLE 6 Adding Polynomials Horizontally
Add.
2x
4
 6 x  7    3 x  5 x  2 
2
4
2
Solution:
 x  x  9
4
2
Slide 5.4-19
EXAMPLE 7 Subtracting Polynomials Horizontally
Perform the subtractions.
7 y
2
 11y  8    3 y  4 y  6 
2
Solution:
  7 y  11y  8    3 y  4 y  6 
2
2
 10 y 2 15 y  2
3
2
2
y

7
y
 4 y  6

from
3
2
14
y

6
y
 2 y  5 .

 14 y 3  6 y 2  2 y  5    2 y 3  7 y 2  4 y  6 
 12 y  y  6 y 11
3
2
Slide 5.4-20
EXAMPLE 8 Subtracting Polynomials Vertically
Subtract.
14 y  6 y  2 y
6
2 y3  7 y 2
3
2
Solution:
14 y3  6 y 2  2 y
3
2
6

2
y

7
y
+
12 y3  y2  2 y  6
Slide 5.4-21
EXAMPLE 9
Adding and Subtracting Polynomials with More Than One Variable
Subtract.
 5m n  3m n
3
2
2
 4mn    7m n  m n  6mn 
3
2
2
Solution:
  5m n  3m n  4mn    7m n  m n  6mn 
3
2
2
3
2
2
 2m3n  4m2 n2  10mn
Slide 5.4-22
EXAMPLE 4 Evaluating a Polynomial
Find the value of 2y3 + 8y − 6 when y = −1.
Solution:
 2  1  8  1  6
3
 2  1  8  6
 2  8  6
 16
Use parentheses around the numbers that are being substituted for the
variable, particularly when substituting a negative number for a
variable that is raised to a power, or a sign error may result.
Slide 5.4-15