Activity Set 3.4CLASS PPTX

Visual Algebra for Teachers

Chapter 3

REAL NUMBERS AND QUADRATIC FUNCTIONS

Visual Algebra for Teachers

Activity Set 3.4

Multiplying and Factoring Polynomials

Visual Algebra for Teachers

PURPOSETo learn: How to find the product of binomials and/or trinomials How to factor some special trinomials

We have already learned how to multiply binomials and factor trinomials of the form using algebra pieces.

In this section, we’ll learn how to do this without the use of manipulatives for some special trinomials

2ax bx c

INTRODUCTION

POLYNOMIAL MULTIPLICATIONThe distributive property (of multiplication over addition) states

or

Multiplication of polynomials is based on the distributive property.

( )a x y ax ay

( )x y b xb yb

Example (polynomial )Distribute the (3x + 7)

Distribute the x and then the 5

Multiply through

Simplify

( 5)(3 7)x x (3 7) 5(3 7)x x x

(3 ) (7) 5(3 ) 5(7)x x x x

23 7 15 35x x x

2( 5)(3 7) 3 8 35x x x x

Classwork (as assigned)

#1 a, b, c, d and e (sharing and extra credit)

#1 f (class discussion)

FACTORING BY GROUPING One technique for factoring quadratics is called factoring by grouping.

There is a visual technique, the “box” method, associated with factoring by grouping.

You can see it is a lot like what we have already done with algebra pieces.

To use this method, factor out any common terms.

FACTORING BY GROUPING Step 1Sketch a blank 2 2 array and add the leading term, ax2, to the top left corner and the constant term, c, to the bottom right corner

FACTORING BY GROUPING Step 2Find two terms that when multiplied together give you ax2 c and when added together, give you the remaining term bx, in your quadratic.

GROUPING EXAMPLEFactor 6x2 + 7x + 2 by grouping

GROUPING EXAMPLEFactor 6x2 + 7x + 2 by grouping

Wee see we need two numbers that add to 7 and multiply to 12.

These must be m = 3 and n = 4 (or visa versa).

FACTORING BY GROUPING Step 3 (with our example: 6x2 + 7x + 2)Factor the common terms out of each row and each column.

FACTORING BY GROUPING Step 4 (with our example: 6x2 + 7x + 2)The sum of the factors for the columns (the top edge value) and the sum of the factors for the rows (the left edge values) are the factors for the quadratic

26 7 2 (2 1)(3 2)x x x x

Classwork (as assigned)

#2 a, b, c, d, e, f, and g (sharing and extra credit)

SPECIAL FACTORIZATIONSTwo important special factorizations are

and the difference of perfect squares

2 2 2( ) 2A B A AB B

2 2( )( )A B A B A B

Classwork (as assigned)

#4 a, b, and c (sharing and extra credit)

Homework

Coursepack: Homework 3.4 (skip #10) Use the box / grouping method

Study for extra credit question on final exam