What is a polynomial?
 Any finite sum of terms.
1) 6x² + 2x – 1
2) 5x – 3
3) 7 x 6  5 x 3  2 x  4
4)  7x 4
5)
15
→ trinomial of degree 2
→ binomial of degree 1
→ polynomial of degree 6
→ monomial of degree 4
→ monomial of degree 1
Adding & Subtracting
 To add or subtract two polynomials, add or subtract
the like terms.
 Like terms
Any two terms with the same variables and exponents
Adding & Subtracting
 Which of the following are like terms?
1) 7xy and 6x
→ not like terms (variables are different)
2) 3y and -4y
→ These are like terms
3) 5x³ and 3x²
→ not like terms (exponents are different)
4) 4x²y³ and -2x³y² → not like terms (exponents)
Adding & Subtracting
 Find the sum:
(5x³ - x + 2x² + 7) + (3x² + 7 – 4x) + (4x² - 8 - x³)
Starting with the highest degree, what are like terms?
= 4x³ + 9x²- 5x + 6
Adding & Subtracting
 Find the sum:
(2x² + x - 5) + (x + x² + 6)
= 3x² + 2x + 1
Adding & Subtracting
 Add the following polynomials:
a) (12x³ + 10) + (18x³ - 3x² + 6)
b) (4x² - 7x + 2) + (-x² + x – 2)
c) (8y³ + 4y² + 3y – 7) + (2y² - 6y + 4)
d) (x³ - 6x) + (2x³ + 9) + (4x² + x³)
(12x³ + 10) + (18x³ - 3x² + 6)
= 30x³ - 3x² + 16
(4x² - 7x + 2) + (-x² + x – 2)
= 3x² - 6x
(8y³ + 4y² + 3y – 7) + (2y² - 6y + 4)
= 8y³ + 6y² - 3y - 3
(x³ - 6x) + (2x³ + 9) + (4x² + x³)
= 4x³ + 4x² - 6x + 9
Adding & Subtracting
 Find the sum:
(2x³ - 3x + 6x² + 7) + (-3x² - 7 + x) + (2x² - x³)
Starting with the highest degree, what are like terms?
= x³ + 5x²- 2x
Subtracting
 So far, the only operation we have used has been
addition.
 Most of the mistakes made on MCAS tests are on
Subtraction problems
Subtraction
 Find the difference
(5x² + 2x – 4) – (3x² - 3x + 6)
You must distribute the negative sign over the
polynomial in the immediately parentheses to the right
5x² + 2x – 4 -3x² + 6x -6 = 2x² + 8x - 10
Subtracting
 Find the difference between the polynomials.
(5h² + 4h + 8) – (3h² + h + 3)
→ Re-write the equation after distributing the minus sign
5h² + 4h + 8 + -3h²- h -3
= 2h² + 3h + 5
Subtracting
(-7z³ + 2z – 7) – (2z³ - z – 3)
-7z³ + 2z - 7 + -2z³ + z + 3
= -9z³ + 3z - 4
Perimeter
 The perimeter of a polygon is the sum of the lengths of
its sides. Write a simplified expression for the
perimeter of this polygon.
x+2
2x + 3
4x – 2
3x + 15
Per. = 2x + 3 + x + 2 + 4x – 2 + 3x + 15
Per. = 10x + 18
Perimeter
 The perimeter of a polygon is the sum of the lengths of
its sides. Write a simplified expression for the
perimeter of this polygon.
x+4
x-3
x-1
x–3
Per. = x - 3 + x + 4 + x – 3 + x - 1
Per. = 4x - 3
So far we have:
 Added polynomials
(4x³ + 2x² + 3) + (2x³ - x² - 2)
6x³ + x² + 1
 Subtracted polynomials
(3x² + 4x + 8) – (2x² + 3x – 5)
3x² + 4x + 8 - 2x² - 3x + 5
x² + x + 13
Multiplying Polynomials
 When we multiply two binomials
(x + 2) (2x – 4)
Use the F.O.I.L. method
F - First
O -Outer
I -Inner
L - Last
F.O.I.L
(x + 2) (2x – 4) = 2x² - 4x + 4x - 8
F - First
x • 2x
O - Outer x • - 4
I - Inner 2 • 2x
L - Last
2 • -4
= 2x² - 8
F.O.I.L
(x + 2) (x – 3) = x² + -3x + 2x + -6
x • x = x²
x • -3 = -3x
2 • x = 2x
2 • -3 = -6
= x²+ -1x - 6
Multiply the following binomials.
a) (2x – 3) (x + 4)
b) (3x + 1) (x + 5)
c) (x – 4) (x + 7)
d) (4x + 4) (2x – 3)
F.O.I.L
(2x - 3) (x + 4) = 2x²+ 8x + -3x+ -12
2x • x = 2x²
2x • 4 = 8x
-3 • x = -3x
-3 • 4 = -12
= 2x² + 5x - 12
F.O.I.L
(3x + 1) (x + 5) = 3x² + 15x + x + 5
3x • x = 3x²
3x • 5 = 15x
1•x =x
1•5 =5
= 3x² + 16x + 5
F.O.I.L
(x - 4) (x + 7) = x² + 7x + -4x- 28
x • x = x²
x • 7 = 7x
-4 • x = -4x
-4 • 7 = -28
= x² + 3x - 28
F.O.I.L
(4x + 4) (2x - 3) = 8x² + -12x + 8x - 12
4x • 2x = 8x²
4x • -3 = -12x
4 • 2x = 8x
4 • -3 = -12
= 8x² - 4x - 12
This week:
 Added polynomials
 Subtracted polynomials
 Multiplied binomials (F.O.I.L.)
Adding Polynomials
(x² + 3x – 4) + (3x² + 5x – 8)
= 4x² + 8x - 12
(2x² – 7) + (4x² + 3x + 3)
= 6x² + 3x - 4
Subtracting Polynomials
(5x² + 2x – 7) - (2x² + 4x – 3)
= 5x² + 2x - 7 - 2x² - 4x + 3
= 3x² - 2x - 4
F.O.I.L
(2x + 3) (4x - 1) = 8x² + -2x + 12x - 3
2x • 4x = 8x²
2x • -1 = -2x
3• 4x = 12x
3 • -1 = -3
= 8x² + 10x - 3