Topic:

Factoring Expressions/Techniques

Hafiz M. Arslan Motor City

Outline:

Some Basic Definitions How to find GCF Factoring by using GCF Factoring by using Grouping Factoring Polynomials by using Difference of Squares Factoring Trinomials by using Perfect Square Factoring Quadratic Factoring by using Cube formula Solving Quadratic by Quadratic Formula Real word problems

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add,subtract,multiply, and divide). Here are some algebraic expressions: 

a + 1

a - b

3x x - a / b

Polynomials

A polynomial is an expression consisting of variables and coefficients which only employs the operations of addition,  subtraction, multiplication, and non-negative integer exponents.

An example of a polynomial of a single variable x is x2 − 4x + 7.

An example in three variables is x3 + 2xyz2 − yz + 1.

Some Basic Definitions

Factors (either numbers or polynomials)When an integer is written as a product of integers, each of the integers in the product is a factor of the original number.When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial.

Factoring

Factoring a polynomial means expressing it as a product of other polynomials.

Greatest Common Factor

Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved.

Finding the GCF of a List of Integers or Terms1) Prime factor the numbers.2) Identify common prime factors.3) Take the product of all common prime factors.

• If there are no common prime factors, GCF is 1.

Greatest Common Factor

Find the GCF of each list of numbers.1) 12 and 8

12 = 2 · 2 · 3 8 = 2 · 2 · 2So the GCF is 2 · 2 = 4.

2) 7 and 20 7 = 1 · 720 = 2 · 2 · 5There are no common prime factors so the GCF is 1.

Example

Greatest Common Factor

1) x3 and x7

x3 = x · x · xx7 = x · x · x · x · x · x · xSo the GCF is x · x · x = x3

2) 6x5 and 4x3

6x5 = 2 · 3 · x · x · x4x3 = 2 · 2 · x · x · x So the GCF is 2 · x · x · x = 2x3

Example

Factoring Method #1

Factoring polynomials with a common monomial

factor (using GCF).

**Always look for a GCF before using any other factoring method.

Steps:

Find the greatest common factor (GCF). Divide the polynomial by the GCF. The

quotient is the other factor. Express the polynomial as the product

of the quotient and the GCF.

Factoring Method #1

3 2 2: 6 12 3Example c d c d cd 3GCF cdStep 1:

Step 2: Divide by GCF

(6c3d 12c2d2 3cd) 3cd 2c2 4cd 1

Factoring Method #1

3cd(2c2 4cd 1)

The answer should look like this:

Ex: 6c3d 12c2d 2 3cd

Factoring Method #1

1. 6x3 3x2 12x

2. 5x2 10x 353. 16x3y4z 8x2 y2z3 12xy3z 2

Factor these on your own looking for a GCF.

Factoring Method #2

Factoring By Groupingfor polynomials

with 4 or more terms

Factoring Method #2

Step 1: Group

3 2

3 4 12b b b Example 1:

b3 3b2 4b 12 Step 2: Factor out GCF from each group

b2 b 3 4 b 3 Step 3: Factor out GCF again b 3 b2 4

Factoring Method #2

3 22 16 8 64x x x

2 x3 8x2 4x 32 2 x3 8x2 4x 32 2 x 2 x 8 4 x 8

Example 2:

Factoring Method #2

2 x 8 x2 4 2 x 8 x 2 x 2

Factor 90 + 15y2 – 18x – 3xy2.

90 + 15y2 – 18x – 3xy2

= 3(30 + 5y2 – 6x – xy2)

= 3(5 · 6 + 5 · y2 – 6 · x – x · y2)

= 3(5(6 + y2) – x (6 + y2)) =

= 3(6 + y2)(5 – x)

Factoring Method #2

Factoring Method #3

Factoring polynomials that are a difference of squares.

To factor, express each term as a square of a monomial then apply the rule...

a2 b2 (a b)(a b)

Factoring Method #3

Ex: x2 16 x2 42 (x 4)(x 4)

Factoring Method #3

Factoring Method #3

Here is another example:

149

x2 81

17

x

2

92 17

x 9

17

x 9

Try these:

Q1: X² -121Q2: 9y²- 169x²Q3: x²- 16

Q: Solve the Word problems by using factoring.1. The area of a square is numerically equal to twice its

perimeter. Find the length of aside of the square.

2. The square of a number equals nine times that number. Find the number.

3. Suppose that four times the square of a number equals 20 times that number. What is the number?

4. The combined area of two squares is 20 square centimeters. Each side of one square is twice as long as a side of the other square. Find the lengths of the sides of each square.

5. The sum of the areas of two squares is 234 square inches. Each side of the larger square is five times the length of aside of the smaller square. Find the length of a side of each square.

Factoring Method #44

Factoring Method #4

Factoring Method #4Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula!

a2 2ab b2 (a b)2

a2 2ab b2 (a b)2

Perfect Squares

Ex: x2 8x 16

2

x 2 4 2

Does the middle term fit the pattern, 2ab?

b

4a

x 8x

x2 8x 16 x 4 2

Yes, the factors are (a + b)2 :

Perfect Squares

Ex: 4x2 12x 9

2x 2 3 2

Does the middle term fit the pattern, 2ab?

b

3a

2x 12x

Perfect Squares

4x 2 12x 9 2x 3 2

Yes, the factors are (a - b)2 :

Perfect Squares

If the x2 term has no coefficient (other than 1)...

Step 1: List all pairs of numbers that multiply to equal the constant, 12.

x2 + 7x + 12

12 = 1 • 12

= 2 • 6

= 3 • 4

Step 2: Choose the pair that adds up to the middle coefficient.

12 = 1 • 12

= 2 • 6

= 3 • 4

Step 3: Fill those numbers into the blanks in the binomials:

( x + )( x + )

x2 + 7x + 12 = ( x + 3)( x + 4)

3 4

Factoring Method #4Factor. x2 + 2x - 24

This time, the constant is negative!

Step 1: List all pairs of numbers that multiply to equal the constant, -24. (To get -24, one number must be positive and one negative.)

-24 = 1 • -24, -1 • 24

= 2 • -12, -2 • 12

= 3 • -8, -3 • 8

= 4 • -6, - 4 • 6

Step 2: Which pair adds up to 2?

Step 3: Write the binomial factors.

x2 + 2x - 24 = ( x - 4)( x + 6)

Factoring Method #4Factor. 3x2 + 14x + 8

This time, the x2 term DOES have a coefficient (other than 1)!

Step 2: List all pairs of numbers that multiply to equal that product, 24.

24 = 1 • 24

= 2 • 12

= 3 • 8

= 4 • 6

Step 3: Which pair adds up to 14?

Step 1: Multiply 3 • 8 = 24 (the leading coefficient & constant).

So then we can write them in the four terms.

3x2 + 12x + 2 x + 8 =( 3x + 2 )( x + 4 )

3x2 + 14x + 8 = (3x + 2)(x + 4)

Special Cases

Sum and Difference of Cubes:

a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2

Special Cases3: 64Example x

(x3 43 ) Rewrite as cubes

Apply the rule for sum of cubes:a3 b3 a b a2 ab b2

(x 4)(x2 4x 16)(x 4)(x2 x 4 42 )

Ex: 8y3 125((2y)3 53)

Apply the rule for difference of cubes:

a3 b3 a b a2 ab b2 2y 5 2y 2 2y5 5 2 2y 5 4y2 10y 25

Quadratic Formula

The quadratic formula is used to solve any quadratic equation.

2 42

x cb b aa

The quadratic formula is:

Standard form of a quadratic equation is: 2 0x xba c

Real word Problem:

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

The Pythagorean Theorem

a2 + b2 = c2

(x + 2)2 + x2 = 202

x2 + 4x + 4 + x2 = 400

2x2 + 4x + 4 = 400

2x2 + 4x – 369 = 02(x2 + 2x – 198) = 0

2(x2 + 2x – 198) = 0

12

1981422 2 x

279242

x

27962

x

20 feet

x + 2

x

20 feet

x + 2

x

2

7962x

2

2.282

22.282

x2

2.282 x

22.26

x

1.13x

22.30

x

1.15xft

20 feet

x + 2

x

1.13x

ft2.28

ft

21.131.132x

28 – 20 = 8 ft

Real Word Problems:

Q 1: A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. When does it hit the ground?

Q 2: A Company is going to make frames as part of a new product they are launching.

The frame will be cut out of a piece of steel, and to keep the weight down, the final area should be 28 cm2.

The inside of the frame has to be 11 cm by 6 cm.What should the width x of the metal be.

Q 3: Two resistors are in parallel, like in this diagram:

The total resistance has been measured at 2 Ohms, and one of the resistors is known to be 3 ohms more than the other.

What are the values of the two resistors?

The formula to work out total resistance "RT" is:

1/RT = 1/R1 + 1/R2

Q 4: An object is thrown downward with an initial velocity of 19 feet per second. The distance, d it travels in an amount of time, t is given by the equation d=19t +. How long does it take the object to fall 50 feet?

Q 5: A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?