Review 10.110.4
Polynomials
Vocabulary
• Monomials - a number, a variable, or a product of a
number and one or more variables. 4x, 20x2yw3, -3,
a2b3, and 3yz are all monomials.
• Polynomials – one or more monomials added or
subtracted
• 4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all polynomials.
Like Terms
Like Terms refers to monomials that have the same
variable(s) but may have different coefficients. The
variables in the terms must have the same powers.
Which terms are like?
3a2b, 4ab2, 3ab, -5ab2
4ab2 and -5ab2 are like.
Even though the others have the same variables, the
exponents are not the same.
3a2b = 3aab, which is different from 4ab2 = 4abb.
Like Terms
Constants are like terms.
Which terms are like?
2x, -3, 5b, 0
-3 and 0 are like.
Which terms are like?
3x, 2x2, 4, x
3x and x are like.
Which terms are like?
2wx, w, 3x, 4xw
2wx and 4xw are like.
Classifying Polynomials
A polynomial with only one term is called a monomial. A polynomial with two terms is
called a binomial. A polynomial with three terms is called a trinomial. Identify the
following polynomials:
6
0
Classified
by degree
constant
–2x
1
linear
monomial
3x + 1
1
linear
binomial
2
quadratic
trinomial
3
cubic
binomial
4
quartic
polynomial
Polynomial
–x 2 + 2x – 5
4x 3 – 8x
2x 4 – 7x 3 – 5x + 1
Degree
Classified by
number of
terms
monomial
Adding Polynomials
Add: (x2 + 3x + 1) + (4x2 +5)
Step 1: Underline like terms:
(x2 + 3x + 1) + (4x2 +5)
Notice: ‘3x’ doesn’t have a like term.
Step 2: Add the coefficients of like terms, do not change
the powers of the variables:
(x2 + 4x2) + 3x + (1 + 5)
5x2 + 3x + 6
Adding Polynomials
Some people prefer to add polynomials by stacking them.
If you choose to do this, be sure to line up the like terms!
(x2 + 3x + 1) + (4x2 +5)
(x2 + 3x + 1)
+ (4x2
+5)
5x2 + 3x + 6
Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)
(2a2 + 3ab + 4b2)
(2a2+3ab+4b2) + (7a2+ab+-2b2)
+ (7a2 + ab + -2b2)
9a2 + 4ab + 2b2
Adding Polynomials
• Add the following polynomials; you may stack them if you
prefer:
1) 3x  7x  3x  4x   6x3  3x
3
3
2) 2w  w  5  4w  7w  1 6w  8w  4
2
2
2
3) 2a  3a  5a  a  4a  3 
3
2
3
3a  3a  9a  3
3
2
Subtracting Polynomials
Subtract: (3x2 + 2x + 7) - (x2 + x + 4)
Step 1: Change subtraction to addition (Keep-Change-Change.).
(3x2 + 2x + 7) + (- x2 + - x + - 4)
Step 2: Underline OR line up the like terms and add.
(3x2 + 2x + 7)
+ (- x2 + - x + - 4)
2x2 + x + 3
Subtracting Polynomials
• Subtract the following polynomials by changing to
addition (Keep-Change-Change.), then add:
1) x  x  4 3x  4x  1 2x  3x  5
2
2
2
2) 9y  3y  1 2y  y  9 7y  4y  10
2
2
2
3) 2g  g  9 g  3g  3  g  g  g  12
2
3
2
3
2
1. Add the following polynomials:
(9y - 7x + 15a) + (-3y + 8x - 8a)
Group your like terms.
9y - 3y - 7x + 8x + 15a - 8a
6y + x + 7a
2. Add the following polynomials:
(3a2 + 3ab - b2) + (4ab + 6b2)
Combine your like terms.
3a2 + 3ab + 4ab - b2 + 6b2
2
2
3a + 7ab + 5b
Add the polynomials.
Y
Y
Y
Y
1.
2.
3.
4.
X
X
1 1 1
X2
+
x2 + 3x + 7y + xy + 8
x2 + 4y + 2x + 3
3x + 7y + 8
x2 + 11xy + 8
Y
Y
Y
X
1 1
1 1 1
XY
3. Add the following polynomials
using column form:
2
2
2
2
(4x - 2xy + 3y ) + (-3x - xy + 2y )
Line up your like terms.
4x2 - 2xy + 3y2
+ -3x2 - xy + 2y2
_________________________
x2 - 3xy + 5y2
4. Subtract the following polynomials:
(9y - 7x + 15a) - (-3y + 8x - 8a)
Rewrite subtraction as adding the
opposite.
(9y - 7x + 15a) + (+ 3y - 8x + 8a)
Group the like terms.
9y + 3y - 7x - 8x + 15a + 8a
12y - 15x + 23a
5. Subtract the following polynomials:
(7a - 10b) - (3a + 4b)
Rewrite subtraction as adding the
opposite.
(7a - 10b) + (- 3a - 4b)
Group the like terms.
7a - 3a - 10b - 4b
4a - 14b
6. Subtract the following polynomials
using column form:
2
2
2
2
(4x - 2xy + 3y ) - (-3x - xy + 2y )
Line up your like terms and add the
opposite.
4x2 - 2xy + 3y2
+ (+ 3x2 + xy - 2y2)
--------------------------------------
7x2 - xy + y2
Find the sum or difference.
(5a – 3b) + (2a + 6b)
1.
2.
3.
4.
3a – 9b
3a + 3b
7a + 3b
7a – 3b
Find the sum or difference.
(5a – 3b) – (2a + 6b)
1.
2.
3.
4.
3a – 9b
3a + 3b
7a + 3b
7a – 9b
Adding Polynomials
Find the sum. Write the answer in standard format.
(5x 3 – x + 2x 2 + 7) + (3x 2 + 7 – 4x) + (4x 2 – 8 – x 3)
SOLUTI
ON
Vertical format: Write each expression in standard form. Align like te
5x 3 + 2x 2 – x + 7
3x 2 – 4x + 7
–+
x 3 + 4x 2
–8
4x 3 + 9x 2 – 5x + 6
Adding Polynomials
Find the sum. Write the answer in standard format.
(2x 2 + x – 5) + (x + x 2 + 6)
SOLUTI
ON
Horizontal format: Add like terms.
(2x 2 + x – 5) + (x + x 2 + 6) = (2x 2 + x 2) + (x + x) + (–5 + 6)
= 3x 2 + 2x + 1
Subtracting Polynomials
Find the difference.
(–2x 3 + 5x 2 – x + 8) – (–2x 2 + 3x – 4)
SOLUTION
Use a vertical format. To subtract, you add the opposite. This
means you multiply each term in the subtracted polynomial by –1
and add.
No change
–2x 3 + 5x 2 – x + 8
–2x 3 + 5x 2 – x + 8
Add the opposite
–2x– 3
+ 3x – 4
2x+3
– 3x + 4
Subtracting Polynomials
Find the difference.
(–2x 3 + 5x 2 – x + 8) – (–2x 2 + 3x – 4)
SOLUTION
Use a vertical format. To subtract, you add the opposite. This
means you multiply each term in the subtracted polynomial by –1
and add.
–2x 3 + 5x 2 – x + 8
–2x 3 + 5x 2 – x + 8
–2x– 3
+ 3x – 4
2x+3
– 3x + 4
5x 2 – 4x + 12
Subtracting Polynomials
Find the difference.
(3x 2 – 5x + 3) – (2x 2 – x – 4)
SOLUTION
Use a horizontal format.
(3x 2 – 5x + 3)=– (3x
(2x22––5x
x –+4)
3) + (–1)(2x 2 – x – 4)
= (3x 2 – 5x + 3) – 2x 2 + x + 4
= (3x 2 – 2x 2) + (– 5x + x) + (3 + 4)
= x 2 – 4x + 7
Multiplying
Polynomials
Distribute and FOIL
Polynomials * Polynomials
Multiplying a Polynomial by another Polynomial requires
more than one distributing step.
Multiply:
(2a + 7b)(3a + 5b)
Distribute 2a(3a + 5b) and distribute 7b(3a + 5b):
6a2 + 10ab
21ab + 35b2
Then add those products, adding like terms:
6a2 + 10ab + 21ab + 35b2 = 6a2 + 31ab + 35b2
Polynomials * Polynomials
An alternative is to stack the polynomials and do long
multiplication.
(2a + 7b)(3a + 5b)
(2a + 7b)
x (3a + 5b)
Multiply by 5b, then by 3a:
When multiplying by 3a, line
up the first term under 3a.
Add like terms:
(2a + 7b)
x (3a + 5b)
21ab + 35b2
+ 6a2 + 10ab
6a2 + 31ab + 35b2
Polynomials * Polynomials
Multiply the following polynomials:
1) x  52x  1
2) 3w  22w  5
3) 2a  a  12a  1
2
2
Polynomials * Polynomials
1) x  52x  1
(x + 5)
x (2x + -1)
-x + -5
+ 2x2 + 10x
2x2 + 9x + -5
2) 3w  22w  5
(3w + -2)
x (2w + -5)
-15w + 10
+ 6w2 + -4w
6w2 + -19w + 10
Polynomials * Polynomials
3) 2a  a  12a  1
2
2
(2a2 + a + -1)
x (2a2 + 1)
2a2 + a + -1
+ 4a4 + 2a3 + -2a2
4a4 + 2a3 + a + -1
Types of Polynomials
• We have names to classify polynomials based on how
many terms they have:
Monomial: a polynomial with one term
Binomial: a polynomial with two terms
Trinomial: a polynomial with three terms
F.O.I.L.
There is an acronym to help us remember how to multiply
two binomials without stacking them.
(2x + -3)(4x + 5)
F : Multiply the First term in each binomial. 2x • 4x = 8x2
O : Multiply the Outer terms in the binomials. 2x • 5 = 10x
I : Multiply the Inner terms in the binomials. -3 • 4x = -12x
L : Multiply the Last term in each binomial. -3 • 5 = -15
(2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15
F.O.I.L.
Use the FOIL method to multiply these binomials:
1) (3a + 4)(2a + 1)
2) (x + 4)(x - 5)
3) (x + 5)(x - 5)
4) (c - 3)(2c - 5)
5) (2w + 3)(2w - 3)
F.O.I.L.
Use the FOIL method to multiply these binomials:
1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4
2) (x + 4)(x - 5) = x2 + -5x + 4x + -20 = x2 + -1x + -20
3) (x + 5)(x - 5) = x2 + -5x + 5x + -25 = x2 + -25
4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c2 + -11c + 15
5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9
There are three techniques you can
use for multiplying polynomials.
The best part about it is that they are all the
same! Huh? Whaddaya mean?
It’s all about how you write it…Here they are!
1)Distributive Property
2)FOIL
3)Box Method
Sit back, relax (but make sure to write this
down), and I’ll show ya!
1) Multiply. (2x + 3)(5x + 8)
Using the distributive property, multiply
2x(5x + 8) + 3(5x + 8).
10x2 + 16x + 15x + 24
Combine like terms.
10x2 + 31x + 24
A shortcut of the distributive property is
called the FOIL method.
The FOIL method is ONLY used when
you multiply 2 binomials. It is an
acronym and tells you which terms to
multiply.
2) Use the FOIL method to multiply the
following binomials:
(y + 3)(y + 7).
(y + 3)(y + 7).
F tells you to multiply the FIRST
terms of each binomial.
y2
(y + 3)(y + 7).
O tells you to multiply the OUTER
terms of each binomial.
y2 + 7y
(y + 3)(y + 7).
I tells you to multiply the INNER
terms of each binomial.
y2 + 7y + 3y
(y + 3)(y + 7).
L tells you to multiply the LAST
terms of each binomial.
y2 + 7y + 3y + 21
Combine like terms.
y2 + 10y + 21
Remember, FOIL reminds you to
multiply the:
First terms
Outer terms
Inner terms
Last terms
The third method is the Box Method.
This method works for every problem!
Here’s how you do it.
Multiply (3x – 5)(5x + 2)
Draw a box. Write a
polynomial on the top and
side of a box. It does not
matter which goes where.
This will be modeled in the
next problem along with
FOIL.
3x
5x
+2
-5
3) Multiply (3x - 5)(5x + 2)
First terms: 15x2
Outer terms: +6x
Inner terms: -25x
Last terms: -10
Combine like terms.
15x2 - 19x – 10
3x
5x
-5
15x2 -25x
+2 +6x
-10
You have 3 techniques. Pick the one you like the best!
4) Multiply (7p - 2)(3p - 4)
First terms: 21p2
Outer terms: -28p
Inner terms: -6p
Last terms: +8
Combine like terms.
21p2 – 34p + 8
7p
3p
-2
21p2 -6p
-4 -28p
+8
Multiply (y + 4)(y – 3)
1.
2.
3.
4.
5.
6.
7.
8.
y2 + y – 12
y2 – y – 12
y2 + 7y – 12
y2 – 7y – 12
y2 + y + 12
y2 – y + 12
y2 + 7y + 12
y2 – 7y + 12
Multiply (2a – 3b)(2a + 4b)
1.
2.
3.
4.
5.
4a2 + 14ab – 12b2
4a2 – 14ab – 12b2
4a2 + 8ab – 6ba – 12b2
4a2 + 2ab – 12b2
4a2 – 2ab – 12b2
5) Multiply (2x - 5)(x2 - 5x + 4)
You cannot use FOIL because they are
not BOTH binomials. You must use the
distributive property.
2x(x2 - 5x + 4) - 5(x2 - 5x + 4)
2x3 - 10x2 + 8x - 5x2 + 25x - 20
Group and combine like terms.
2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20
5) Multiply (2x - 5)(x2 - 5x + 4)
You cannot use FOIL because they are not BOTH
binomials. You must use the distributive property or
box method.
2x
-5
x2
-5x
+4
2x3
-10x2
+8x
-5x2 +25x
-20
Almost
done!
Go to
the next
slide!
5) Multiply (2x - 5)(x2 - 5x + 4)
Combine like terms!
x2
-5x
+4
2x
2x3
-10x2
+8x
-5
-5x2 +25x
-20
2x3 – 15x2 + 33x - 20
Multiply (2p + 1)(p2 – 3p + 4)
1.
2.
3.
4.
2p3 + 2p3 + p + 4
y2 – y – 12
y2 + 7y – 12
y2 – 7y – 12
Example:
(x – 6)(2x + 1)
x(2x) + x(1) – (6)2x – 6(1)
2x2 + x – 12x – 6
2x2 – 11x – 6
2x2(3xy + 7x – 2y)
2x2(3xy + 7x – 2y)
2x2(3xy) + 2x2(7x) + 2x2(–2y)
6x3y + 14x2 – 4x2y
(x + 4)(x – 3)
(x + 4)(x – 3)
x(x) + x(–3) + 4(x) + 4(–3)
x2 – 3x + 4x – 12
x2 + x – 12
(2y – 3x)(y – 2)
(2y – 3x)(y – 2)
2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2)
2y2 – 4y – 3xy + 6x
There are formulas (shortcuts) that
work for certain polynomial
multiplication problems.
2
b)
2
a
2
b
(a +
= + 2ab +
(a - b)2 = a2 – 2ab + b2
2
2
(a - b)(a + b) = a - b
Being able to use these formulas will help you in
the future when you have to factor. If you do not
remember the formulas, you can always multiply
using distributive, FOIL, or the box method.
Let’s try one!
1) Multiply: (x + 4)2
You can multiply this by rewriting this as
(x + 4)(x + 4)
OR
You can use the following rule as a shortcut:
(a + b)2 = a2 + 2ab + b2
For comparison, I’ll show you both ways.
1) Multiply (x + 4)(x + 4)
First terms: x2
Outer terms: +4x
Inner terms: +4x
Last terms: +16
Combine like terms.
x2 +8x + 16
Notice you
have two of
the same
answer?
x
x
+4
x2
+4x
+4 +4x +16
Now let’s do it with the shortcut!
1) Multiply: (x + 4)2
That’s why
the 2 is in
the formula!
using (a + b)2 = a2 + 2ab + b2
a is the first term, b is the second term
(x + 4)2
a = x and b = 4
Plug into the formula
a2 + 2ab + b2
(x)2 + 2(x)(4) + (4)2
This is the
Simplify.
same answer!
x2 + 8x+ 16
2) Multiply: (3x + 2y)2
using (a + b)2 = a2 + 2ab + b2
(3x + 2y)2
a = 3x and b = 2y
Plug into the formula
a2 + 2ab + b2
(3x)2 + 2(3x)(2y) + (2y)2
Simplify
9x2 + 12xy +4y2
Multiply (2a + 3)2
1.
2.
3.
4.
4a2 – 9
4a2 + 9
4a2 + 36a + 9
4a2 + 12a + 9
Multiply: (x – 5)2
using (a – b)2 = a2 – 2ab + b2
Everything is the same except the signs!
(x)2 – 2(x)(5) + (5)2
x2 – 10x + 25
4) Multiply: (4x – y)2
(4x)2 – 2(4x)(y) + (y)2
16x2 – 8xy + y2
Multiply (x – y)2
1.
2.
3.
4.
x2 + 2xy + y2
x2 – 2xy + y2
x2 + y2
x2 – y2
5) Multiply (x – 3)(x + 3)
First terms: x2
Outer terms: +3x
Inner terms: -3x
Last terms: -9
Combine like terms.
x2 – 9
Notice the
middle terms
eliminate
each other!
x
-3
x2
-3x
+3 +3x
-9
x
This is called the difference of squares.
5) Multiply (x – 3)(x + 3) using
2
2
(a – b)(a + b) = a – b
You can only use this rule when the binomials
are exactly the same except for the sign.
(x – 3)(x + 3)
a = x and b = 3
(x)2 – (3)2
x2 – 9
6) Multiply: (y – 2)(y + 2)
(y)2 – (2)2
y2 – 4
7) Multiply: (5a + 6b)(5a – 6b)
(5a)2 – (6b)2
25a2 – 36b2
Multiply (4m – 3n)(4m + 3n)
1.
2.
3.
4.
16m2 – 9n2
16m2 + 9n2
16m2 – 24mn - 9n2
16m2 + 24mn + 9n2
Simplify.
1) (x  5) 2
(x  5)(x  5)
2
x  10x  25
2
(m

2)
2)
(m  2)(m  2)
2
m  4m  4
Follow the pattern!
(a  b)  a  2ab  b
2
2
(x  5)  x  10x  25
2
2

y
 6y  9
(y  3)
2
Last
Term
2
Twice
the
Last
Term
2
Square
of the
Last
Term
Difference of Squares.
Multiply.
2
(x

3)(x

3)
1)
 x 9
2
(m

7)(m

7)
2)
 m  49
3) (y  10)(y  10)  y  100
2
2
(t

8)(t

8)
4)
 t  64
Inner and Outer terms cancel!
Example 2: Finding Products in the Form (a – b)2
Multiply.
A. (x – 6)2
(a – b) = a2 – 2ab + b2
(x – 6) = x2 – 2x(6) + (6)2
= x – 12x + 36
B. (4m – 10)2
Use the rule for (a – b)2.
Identify a and b: a = x and b
= 6.
Simplify.
Example 2: Finding Products in the Form (a – b)2
Multiply.
C. (2x – 5y )2
D. (7 – r3)2
Check It Out! Example 2
Multiply.
a. (x – 7)2
b. (3b – 2c)2
Check It Out! Example 2c
Multiply.
(a2 – 4)2
(a + b)(a – b) = a2 – b2
A binomial of the form a2 – b2 is called a difference
of two squares.
Example 3: Finding Products in the Form (a + b)(a – b)
Multiply.
A. (x + 4)(x – 4)
(a + b)(a – b) = a2 – b2
(x + 4)(x – 4) = x2 – 42
= x2 – 16
B. (p2 + 8q)(p2 – 8q)
Use the rule for (a + b)(a – b).
Identify a and b: a = x and
b = 4.
Simplify.
Example 3: Finding Products in the Form (a + b)(a – b)
Multiply.
C. (10 + b)(10 – b)
Check It Out! Example 3
Multiply.
a. (x + 8)(x – 8)
b. (3 + 2y2)(3 – 2y2)
Check It Out! Example 3
Multiply.
c. (9 + r)(9 – r)