factoring polynomials

Factoring PolynomialsGeogebra Efil Mileny Catayong

Common Monomial

Factor

Factoring the GCF from Polynomials

ReviewAlgebraic Factorization is the writing of an expression as the product of prime numbers and variables with no variables having an exponent greater than one.Example:14x2y = 2 7 x x y

Factoring the GCF from Polynomials

1. Write each term on a separate line and then write the algebraic factorization of each term.

2. “Pair up” all factors that occur in each term and circle them.

3. Multiply what is circled. This is the GCF. When writing the answer, put it outside parenthesis. The “left-overs” for each term should be multiplied and put inside parenthesis.

Example: FACTOR: 3x3 + 6x2y – 15xy2

Step 1

3x3 = 3 x x x 6x2y = 2 3 x x y 15xy2 = 3 5 x y y

Step 3

Step 4

3x3 = 3 x x x 6x2y = 2 3 x x y 15xy2 = 3 5 x y

y

3x (x2 + 2xy – 5y2)

Try these yourself.

1. 15x2 – 20xy

2. 3x2 + 15x

3. 20abc + 15a2c – 5ac

Factoring Trinomials

Multiply. (x+3)(x+2)

Review

Multiplying Binomials (FOIL)

x • x + x • 2 + 3 • x + 3 • 2

F O I L

= x2+ 2x + 3x + 6

= x2+ 5x + 6

Distribute.

x + 3

x

+

2

Using Algebra Tiles, we have:

= x2 + 5x + 6

Review

Multiplying Binomials (Tiles)Multiply. (x+3)(x+2)

x2 x

x 1

x x

x1 1

1 1 1

How can we factor trinomials such as x2 + 7x + 12 back into binomials?

One method is to again use algebra tiles:

1) Start with x2.

Factoring Trinomials (Tiles)

2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles.

x2 x x xxx

x

x

1 1 1

1 1 1

1 1

1 1

1 1

1) Start with x2.

Factoring Trinomials (Tiles)

2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles.

x2 x x xxx

x 1 1 1

1 1 1

1 1

1 1 1

1

3) Rearrange the tiles until they form a rectangle!

Still not a rectangle.

x

1) Start with x2.

Factoring Trinomials (Tiles)

2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles.

x2 x x xx

x 1 1 1

1 1 1

1

1

1

1 11

3) Rearrange the tiles until they form a rectangle!

A rectangle!!!

x

x

4) Top factor:The # of x2 tiles = x’sThe # of “x” and “1” columns = constant.

Factoring Trinomials (Tiles)

5) Side factor:The # of x2 tiles = x’sThe # of “x” and “1” rows = constant.

x2 x x xx

x 1 1 1

1 1 1

1

1

1

1 11

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

x

x

x + 4

x

+

3

Again, we will factor trinomials such as x2 + 7x + 12 back into binomials.

Factoring Trinomials (Method 2)

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.

x2 + 7x + 12

12 = 1 • 12

= 2 • 6

= 3 • 4Step 3: Fill those numbers into the blanks in the binomials:

( x + )( x + )3 4

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

Factor each trinomial, if possible. The first four do NOT have leading coefficients, the last two DO have leading coefficients. Watch out for signs!!

1) t2 – 4t – 21

2) x2 + 12x + 32

3) x2 –10x + 24

4) x2 + 3x – 18

5) 2x2 + x – 21

6) 3x2 + 11x + 10

Factor These Trinomials!

Perfect Square Trinomials

When factoring using perfect square trinomials, look for the following three things:–3 terms– last term must be positive–first and last terms must be perfect

squares If all three of the above are true, write

one ( )2 using the sign of the middle term.

Try These1. a2 – 8a + 162. x2 + 10x + 253. 4y2 + 16y + 164. 9y2 + 30y + 255. 3r2 – 18r + 276. 2a2 + 8a - 8

Difference of Squares

Difference of Squares

When factoring using a difference of squares, look for the following three things:–only 2 terms–minus sign between them–both terms must be perfect

squares

If all 3 of the above are true, write two

( ), one with a + sign and one with a – sign : ( + ) ( - ).

Difference of Squares

Try These1. a2 – 8a + 162. x2 + 10x + 253. 4y2 + 16y + 164. 9y2 + 30y + 255. 3r2 – 18r + 276. 2a2 + 8a - 8

Factoring Four Term Polynomials

Factor by Grouping

When polynomials contain four terms, it is sometimes easier to group like terms in order to factor.

Your goal is to create a common factor. You can also move terms around in the

polynomial to create a common factor. Practice makes you better in

recognizing common factors.

Factor by GroupingExample

FACTOR: 3xy - 21y + 5x – 35 Factor the first two terms: 3xy - 21y = 3y (x – 7) Factor the last two terms: + 5x - 35 = 5 (x – 7) The white parentheses are the

same so it’s the common factor Now you have a common factor

(x - 7) (3y + 5)

Factoring Completely

Factoring Completely Now that we’ve learned all the

types of factoring, we need to remember to use them all.

Whenever it says to factor, you must break down the expression into the smallest possible

factors.

Let’s review all the ways to factor.