Polynomials and Factoring

1
Polynomials and Factoring
2
Aim 9-1 How do we add and subtract polynomials?

• Monomial is an expression that is a number, a
  variable or a product of a number.
• Ex. 7x2y5 Degree 7 (Add the exponents 2 5 7)
• Degree of a polynomial is the sum of the
  exponents of its variable .
• Ex. 3x4 5x2 - 7x 1 The degree is 4.
• (Look for the highest exponent to identify the
  degree of a polynomial)

3
What is a polynomial?

• Polynomial is a monomial or the sum or difference
  of two or more monomials.

• Standard form of a polynomial means that the
  degree of the monomial terms decrease from left
  to right.
• Ex 2x3 x2 x 3

4
Classifying Polynomials

• Name the polynomial based on its degree and the
  number of its terms.
• -2x 5

5
Classifying Polynomials

• Name the polynomial based on its degree and the
  number of its terms.
• -2x 5
• Answer Linear binomial

6
Classifying Polynomials

• Write each in standard form. Then name the
  polynomial based on its degree and the number of
  its terms.
• 3x4 4 2x2 5x4
• Answer 8x4 2x2 4
• fourth degree trinomial

7
Practice

• Write each in standard form. Then name the
  polynomial based on its degree and the number of
  its terms.
• 6x2 7 9x4
• 3y 4 y3
• 8 7v 11v

8
Adding Polynomials

• (4x2 6x 7) ( 2x2 9x 1)
• Hint Combine like terms.
• Answer 6x2 3x 8

9
Practice

• Simplify each sum.
• (12m2 4) (8m2 5)
• (t2 6) ( 3t2 11)
• (2p3 6p2 10p) (9p3 11p2 3p)

10
Subtracting Polynomials

• (2x3 5x2 3x ) ( x3 8x2 11)
• The signs of the second polynomial change to
  their opposite.
• (2x3 5x2 3x ) ( - x3 8x2 - 11)
• Now you can add the expressions by combining like
  terms
• .
• Answer x3 13x2 3x - 11

11
Summary

• Write a polynomial and identify the following
• The degree
• Number of terms
• Explain how to add and subtract polynomials.

12
Aim 9-2 How do we multiply a polynomial by a
monomial?

• -4y2( 5y4 3y2 2)
• Multiply the 4y2 with each term inside the
  parenthesis.
• Answer -20 y6 12 y4 8 y2

13
Finding the greatest common factor

• 4x3 12x2 - 8x
• (Whats the GCF of 4, 12, and 8?
• Whats the GCF of x3, x2, and x?)
• The GCF is 4x.

14
Practice

• Find the GCF of the terms of each polynomial.
• 5v5 10v3
• 3t2 18
• 4b3 2b2 6b

15
Factoring Out a Monomial

• 3x3 12x2 15x
• Think What is the GCF of each term?
• GCF is 3x.
• Answer 3x (x2 4x 5)

16
Practice

• Factor Out a Monomial. Use the GCF to factor each
  monomial.
• 8x2 12
• 5d3 10d
• 6m3 12m2 24m

17
Summary

• Explain how do you find the GCF of a polynomial.

18
Do Now

• Take out your homework and unit plan.
• Complete p. 464 prob. 25, 27-37(odd), 40
• P. 465 prob. 45-48

19
Aim 9-3 How do we multiply binomials using FOIL?

• Strategy 1 Using the distributive property
• ( x 4) (2x 3) Distribute x 4.
• 2x (x 4) 3 (x 4) Then simplify.
• 2x2 8x 3x 12 Combine like terms.
• Answer 2x2 11x 12

20
Using the Distributive Property

• Simplify each product.
• 1. (6h 7 ) ( 2h 3 )
• 2. (5m 2) ( 8m 1)

21

• Using the FOIL method
• F- First terms
• O- outer terms
• I- Inner terms
• L- last terms
• Example (3x 5) (2x 7)
• F O I L
• 3x 2x 3x 7 (-5) 2x (-5)7
• Then simplify.
• 6x2 21x -10x -35 6x2 11x - 35

22
Simplify using FOIL.

• (3x 4) (2x 5)
• (3x 4) (2x 5)

23
Applying Multiplication of Polynomials

• Suppose you have a rectangle with dimensions 2x
  5 and 3x 1 and inside you have a smaller
  rectangle with the dimensions x 2 and x. What
  is the area of the unshaded region?

24
Multiplying a Trinomial and a Binomial

• (4x2 x 6) (2x 3)
• You may use the distributive property.
• OR you may use the vertical method.
• 4x2 x 6
• 2x 3
• -12x2-3x 18
• 8x32x2 -12x 0
• 8x3-10x2 -15x 18

25
Simplify

• (6n 8 ) (2n2 n 7)

26
Summary

• What do the letters in FOIL represent?