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·xx7

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 x5

x3 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 x5

x3 x3

x·x·x·x·x x5-3

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 x5

x3 x3

x·x·x·x·x x5-3

x·x·x x2 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 OR (y6)3

(y6)(y6)(y6)

y6+6+6

y18

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6

y18

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

(7·4)

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

(7·4)

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

(7·4)(x3·x)

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

(7·4)(x3·x)

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

(7·4)(x3·x)(y3·y-5)

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

(7·4)(x3·x)(y3·y-5) 28x4y-2

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3)

(7·4)(x3·x)(y3·y-5) 28x4y-2

28x4

y2

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

Ex. 4 Simplify (7x3y-5)(4xy3) (7·4)(x3·x)(y3·y-5) 28x4y-2

28x4

y2

Ex. 5 Simplify -5x3y3z4

20x3y7z4

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

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

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

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

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

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

Ex. 3 Simplify (y6)3 OR (y6)3

(y6)(y6)(y6) y6·3

y6+6+6 y18

y18

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

• 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