5.3: Add, Subtract, & Multiply Polynomials

5.3: Add, Subtract, & Multiply Polynomials
1.
Objectives:
To find the sum,
difference, and
product of
polynomials
Assignment:
• P. 349-352: 1, 2, 3-51
M3, 56, 60, 65, 66
• Binomial Theorem
Presentation and
Supplement
Warm-Up
Pascal’s Triangle
appears at the right.
Find the pattern and
generate the next
two rows.
1
1
2
1
1
1
4
1
3
3
1
1
1
6
1
4
1
5 10 10 5 1
6 15 20 15 6 1
Objective 1
Polynomial
What makes one of these things a
polynomial?
Polynomial
NOT a Polynomial
2𝑥 2
3𝑥𝑦
𝑥 4 − 81
2𝑥 − 𝑥
𝑥 2 − 3𝑥 + 4
𝑥 −1 + 2𝑥 −2 − 4𝑥 −3
Polynomial
A polynomial in x is an expression of the
form:
an x n 
 a2 x 2  a1 x  a0
Exponents:
All exponents
are
nonnegative
integers
The degree
of the
polynomial is
n, as long as
an  0
The degree is
the highest
power of the
polynomial
Polynomial
A polynomial in x is an expression of the
form:
an x n 
 a2 x 2  a1 x  a0
Coefficients:
All
coefficients ak
are real
numbers
an is the
leading
coefficient
a0 is the
constant term
Polynomial
A polynomial in x is an expression of the
form:
an x n 
 a2 x 2  a1 x  a0
Standard form:
Variables written with powers in
descending order
Polynomial
A polynomial in x is an expression of the
form:
an x n 
 a2 x 2  a1 x  a0
Monomial
Binomial
Trinomial
One term
Two terms
Three terms
2𝑥 2
𝑥 4 − 81
𝑥 2 − 3𝑥 + 4
Types and Degrees of Polys
Degree
Type
Standard Form
0
Constant
f(x) = a0
1
Linear
f(x) = a1x + a0
2
Quadratic
f(x) = a2x2 + a1x + a0
3
Cubic
f(x) = a3x3 + a2x2 + a1x + a0
4
Quartic
f(x) = a4x4 + a3x3 + a2x2 + a1x + a0
5
Quintic
f(x) = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
6
Sextic
f(x) = a6x6 + a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
Types and Degrees of Polys
Degree
Type
Examples
0
Constant
f(x) = 42
1
Linear
f(x) = 3x – 7
2
Quadratic
f(x) = -1.5x2 + .75x + 6
3
Cubic
f(x) = x3 + 4x2 – 5x + 1
4
Quartic
f(x) = 7x4 + 9.5x3 +2x2 + 10x – 1
5
Quintic
f(x) = 4x5 + 8x4 + 15x3 + 16x2 + 23x + 42
6
Sextic
f(x) = x6 – 2x5 + 3x4 – 4x3 + 5x2 – 6x + 7
Objective 1
Like Terms
Like terms are simply monomials with the
same variable raised to the same power
2𝑥 3 + 𝑥 2 − 4𝑥 3 + 5𝑥 2 − 𝑥 − 7
To add and subtract
polynomials, just add or
subtract the coefficients of
like terms
The
powers DO
NOT
change!
Exercise 1
Add using a horizontal format.
 4x
3
 
 4 x 2  3x  10  5x3  2 x 2  4 x  4
 x3 2x 2 7x 6
It’s often helpful to underline like terms
the same number of times.

Exercise 2
Add using a vertical format.
 2x
3
 
 2 x 2  3x  5  3x3  4 x 2  x  7

2 x3
2 x 2
3x
3x3
5x 3
4 x 2
2x 2
x
7
4x 2

5
Align like terms and add the old-fashioned way.
Distributive Property
Distributive Property
When subtracting polynomials, you have to
distribute the negative sign:



 3x3  4 x 2  x  7  1 3x3  4 x 2  x  7

 3x3 4x 2  x 7
Exercise 3
Subtract 6y2 – 6y – 13 from 3y2 – 4y + 7 in a
horizontal format.

 
3 y 2  4 y  7  6 y 2  6 y  13
 3 y 2  4 y  7  6 y 2  6 y  13
 3y 2 2 y 20

Exercise 4
Subtract –4x3 + 6x2 + 9x – 3 from
3x3 + 4x2 + 7x + 12 in a vertical format.
3x3
4 x 2
7 x 12
 (4 x3
6 x 2
9 x
3x3
4 x 2
7 x 12
3)  4 x3 6 x 2 9 x 3
7x3 2x 2 2x 15
Basically you have to turn a subtraction
problem into addition by adding the
opposite.
Exercise 5
Find the sum or difference.
1. (t2 – 6t + 2) + (5t2 – t – 8)
2. (8d – 3 + 9d3) – (d3 – 13d2 – 4)
Objective 1
Polynomial Multiplication
Polynomial multiplication is basically
repeated application of the distributive
property.
 6x  3x2  4x 1
    6x  4x    6x  1
  6 x  3x 2
 18x3
24x 2
6x
Multiply coefficients and add exponents
Polynomial Multiplication
Polynomial multiplication is basically
repeated application of the distributive
property.
 2 x  3 5x  1
Product of the
First terms
 10x 2 2x 15x 3
Product of the
Outside terms
Product of the
Inside terms
 10 x2  13x  3
Product of the
Last terms
Exercise 6
Find the product.
 a  b  a  b 
 a 2  b2
Difference of two squares
Protip: Difference of 2 Squares
When finding the difference of 2 squares,
 a  b  a  b 
 a 2  b2
Square the first term
Square the second term
just square the first number, square the
second number, and take the difference.
The middle term cancels out.
Exercise 7
1. (a + b)2
 a 2  2ab  b2
2. (a – b)2
 a 2  2ab  b2
Perfect square trinomials
Protip: Perfect Square Trinomials
When expanding the square of a binomial:
1. (a + b)2  a 2  2ab  b2
Square the
first term
The middle term gets
the sign between the
original terms
Square the
second term
Multiply both terms
and then double
Protip: Perfect Square Trinomials
When expanding the square of a binomial:
2. (a − b)2  a 2  2ab  b 2
Square the
first term
The middle term gets
the sign between the
original terms
Square the
second term
Multiply both terms
and then double
Exercise 8
1. (5y – 3)(5y + 3)
2. (4a + 7)2
3. (2x – 3)2
Objective 1
Polynomial Multiplication
When multiplying a polynomial by a
polynomial, each term of the first
polynomial must be multiplied by each term
of the second polynomial.
Again, this is just the distributive
property used multiple times.
Exercise 9
Multiply x2 – 2x + 3 and x + 5 in a horizontal
format.
 x  5   x 2  2 x  3
 x3 2x 2 3x 5x 2 10x 15
 x3  3x 2  7 x  15
Exercise 10
Multiply 3x2 + 3x + 5 and 2x + 3 in a vertical
format.
3x 2
3 x
9x 2
2 x 3
9x 15

5
Exercise 10
Multiply 3x2 + 3x + 5 and 2x + 3 in a vertical
format.
3x 2

3 x
5
2 x 3
9x 2 9x 15
6x 3 6x 2 10x
6x 3 15x 2 19x 15
Exercise 11
Find the product.
1. (x + 2)(3x2 – x – 5)
2. (x – 3)(4 + 2x – x2)
Exercise 12
1.
(a + b)3   a  b  a  b 
2


  a  b  a 2  2ab  b 2
 a3  2a 2b  ab2  a 2b  2ab2  b3
 a3  3a 2b  3ab2  b3
Cube of a Binomial
2.
(a – b)3   a  b  a  b 2


  a  b  a 2  2ab  b2
 a3  2a 2b  ab2  a 2b  2ab2  b3
 a3  3a 2b  3ab2  b3
Protip: Cube of a Binomial
To cube a binomial,
just use Pascal’s
Triangle.
1
1
2
1
The first term
starts to the 3rd
power, and
then decreases
the power by
one each term
1
1
3
1
3
1
 a  b   1a3  3a 2b  3ab2  1b3
3
Protip: Cube of a Binomial
To cube a binomial,
just use Pascal’s
Triangle.
1
1
2
1
The second
term starts to
the 0 power, and
then increases
the power by
one each term
1
1
3
1
3
1
 a  b   1a3  3a 2b  3ab2  1b3
3
Protip: Cube of a Binomial
To cube a binomial,
just use Pascal’s
Triangle.
1
1
1
If the sign in the
original binomial
is −, alternate
the signs in the
answer: + − + −
1
2
+ 1 − 3 +3
1
−1
 a  b   1a 3  3a 2b  3ab 2  1b3
3
Notice that this pattern would
work on squaring binomials
Exercise 13
1. (x + 4)3
2. (mn – 6)3
Exercise 14
Find the product.
(a – 5)(a + 2)(a + 6)
A rectangular box has
sides whose lengths
(in inches) are (x + 2),
(x – 2), and (2x + 1).
Write a polynomial
model in standard
form for the volume of
the box. Find the
volume when x = 5.
2x + 1
Exercise 15
x+2
2x + 1
Exercise 15
x+2
5.3: Add, Subtract, & Multiply Polynomials
1.
Objectives:
To find the sum,
difference, and
product of
polynomials
1
1
2
1
1
1
4
1
3
3
1
1
1
6
1
4
1
5 10 10 5 1
6 15 20 15 6 1
Assignment
• P. 349-352: 1, 2,
3-51 M3, 56, 60,
65, 66
• Binomial Theorem
Presentation and
Supplement
“Who’s Paul E. Nomeal?”