Download 5.1 - Monomials

6.1 - Polynomials
• Monomial
• Monomial – 1 term
• Monomial – 1 term; a variable
• Monomial – 1 term; a variable, number
times a variable
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
x7
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
x7
Ex. 2 Simplify x5
x3
• Monomial – 1 term; a variable, number
times a variable, or multiple of numbers
and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
x7
Ex. 2 Simplify x5
x3
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
x·x·x·x·x·x·x
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
x2
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
OR x4·x3
x·x·x·x·x·x·x
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
x2
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
OR x4·x3
x·x·x·x·x·x·x
x4+3
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
x2
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
OR x4·x3
x·x·x·x·x·x·x
x4+3
x7
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
x2
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
OR x4·x3
x·x·x·x·x·x·x
x4+3
x7
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
x2
x5
x3
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
OR x4·x3
x·x·x·x·x·x·x
x4+3
x7
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
x2
x5
x3
x5-3
• Monomial – 1 term; a variable, number times a
variable, or multiple of numbers and variables.
Operations with Monomials:
Ex. 1 Simplify x4·x3
OR x4·x3
x·x·x·x·x·x·x
x4+3
x7
x7
Ex. 2 Simplify x5
x3
x·x·x·x·x
x·x·x
x2
x5
x3
x5-3
x2
Ex. 3 Simplify (y6)3
Ex. 3 Simplify (y6)3
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
(y6)3
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
(y6)3
y6·3
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
28x4y-2
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
28x4y-2
28x4
y2
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
28x4y-2
28x4
y2
Ex. 5 Simplify -5x3y3z4
20x3y7z4
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
28x4y-2
28x4
y2
Ex. 5 Simplify -5x3y3z4
20x3y7z4
-5 · x3-3 · y3-7 · z4-4
20
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
28x4y-2
28x4
y2
Ex. 5 Simplify -5x3y3z4
20x3y7z4
-5 · x3-3 · y3-7 · z4-4
20
-¼·x0·y-4·z0
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
28x4y-2
28x4
y2
Ex. 5 Simplify -5x3y3z4
20x3y7z4
-5 · x3-3 · y3-7 · z4-4
20
-¼·x0·y-4·z0
-¼·1·y-4·1
(y6)3
y6·3
y18
Ex. 3 Simplify (y6)3
(y6)(y6)(y6)
y6+6+6
y18
OR
Ex. 4 Simplify (7x3y-5)(4xy3)
(7·4)(x3·x)(y3·y-5)
28x4y-2
28x4
y2
Ex. 5 Simplify -5x3y3z4
20x3y7z4
-5 · x3-3 · y3-7 · z4-4
20
-¼·x0·y-4·z0
-¼·1·y-4·1
-1
4y4
(y6)3
y6·3
y18
• Polynomials
• Polynomials – Expressions with more than
1 term.
• Polynomials – Expressions with more than
1 term.
• Degree
• Polynomials – Expressions with more than
1 term.
• Degree – highest combination of
exponents on single term.
• Polynomials – Expressions with more than
1 term.
• Degree – highest combination of
exponents on single term.
Ex. 1 Determine the degree of the
polynomial.
• Polynomials – Expressions with more than
1 term.
• Degree – highest combination of
exponents on single term.
Ex. 1 Determine the degree of the
polynomial.
x3y5 – 9x4
• Polynomials – Expressions with more than
1 term.
• Degree – highest combination of
exponents on single term.
Ex. 1 Determine the degree of the
polynomial.
x3y5 – 9x4
• Polynomials – Expressions with more than
1 term.
• Degree – highest combination of
exponents on single term.
Ex. 1 Determine the degree of the
polynomial.
x3y5 – 9x4
degree = 8
Operations with Polynomials:
Operations with Polynomials:
Ex. 1 Simplify (5y + 3y2) + (-8y - 6y2)
Operations with Polynomials:
Ex. 1 Simplify (5y + 3y2) + (-8y - 6y2)
5y - 8y + 3y2 - 6y2
Operations with Polynomials:
Ex. 1 Simplify (5y + 3y2) + (-8y - 6y2)
5y - 8y + 3y2 - 6y2
-3y - 3y2
Operations with Polynomials:
Ex. 1 Simplify (5y + 3y2) + (-8y - 6y2)
5y - 8y + 3y2 - 6y2
-3y - 3y2
Ex. 2 Simplify (9r2 + 6r + 16) – (8r2 – 7r +
10)
Operations with Polynomials:
Ex. 1 Simplify (5y + 3y2) + (-8y - 6y2)
5y - 8y + 3y2 - 6y2
-3y - 3y2
Ex. 2 Simplify (9r2 + 6r + 16) – (8r2 – 7r +
10)
(9r2 + 6r + 16) + (-8r2 + 7r – 10)
Operations with Polynomials:
Ex. 1 Simplify (5y + 3y2) + (-8y - 6y2)
5y - 8y + 3y2 - 6y2
-3y - 3y2
Ex. 2 Simplify (9r2 + 6r + 16) – (8r2 – 7r +
10)
(9r2 + 6r + 16) + (-8r2 + 7r – 10)
9r2 – 8r2 + 6r + 7r + 16 – 10
Operations with Polynomials:
Ex. 1 Simplify (5y + 3y2) + (-8y - 6y2)
5y - 8y + 3y2 - 6y2
-3y - 3y2
Ex. 2 Simplify (9r2 + 6r + 16) – (8r2 – 7r +
10)
(9r2 + 6r + 16) + (-8r2 + 7r – 10)
9r2 – 8r2 + 6r + 7r + 16 – 10
r2 + 13r + 6
Ex. 3 Simplify 4b(cb – zd)
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
4b2c – 4bdz
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
4b2c – 4bdz
Ex. 4 Simplify (3x + 8)(2x + 6)
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
4b2c – 4bdz
Ex. 4 Simplify (3x + 8)(2x + 6)
3x(2x + 6) + 8(2x + 6)
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
4b2c – 4bdz
Ex. 4 Simplify (3x + 8)(2x + 6)
3x(2x + 6) + 8(2x + 6)
3x·2x + 3x·6 + 8·2x + 8·6
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
4b2c – 4bdz
Ex. 4 Simplify (3x + 8)(2x + 6)
3x(2x + 6) + 8(2x + 6)
3x·2x + 3x·6 + 8·2x + 8·6
3·2·x·x + 3·6·x + 8·2·x + 8·6
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
4b2c – 4bdz
Ex. 4 Simplify (3x + 8)(2x + 6)
3x(2x + 6) + 8(2x + 6)
3x·2x + 3x·6 + 8·2x + 8·6
3·2·x·x + 3·6·x + 8·2·x + 8·6
6x2 + 18x + 16x + 48
Ex. 3 Simplify 4b(cb – zd)
4b·cb – 4b·zd
4b2c – 4bdz
Ex. 4 Simplify (3x + 8)(2x + 6)
3x(2x + 6) + 8(2x + 6)
3x·2x + 3x·6 + 8·2x + 8·6
3·2·x·x + 3·6·x + 8·2·x + 8·6
6x2 + 18x + 16x + 48
6x2 + 34x + 48