Factoring Done by:Rashed salmeen Grade:9ASP2 Prime factorization - - PowerPoint PPT Presentation

Factoring

Done by:Rashed salmeen Grade:9ASP2

Prime factorization

Prime factorization:is finding which prime numbers multiply together to make the original number.

Examples:

3 9 2

18

3

27

3 9 18=2x3x3 3 3 27=3x3x3

greatest common factors(GCF)

Greatest common factors(GCF):the greatest number that is a factor of two given number

Example:

20

10 2 5 2 20=2x2x5 36 2 18 9 2 3 3 36=2x2x3x3 GCF=2x2=4

Factoring by grouping

First step find the GCF

12a3-9a2 +20a-15 GCF:1

Separate the four terms to two binomial Factor out the GCF of each group

GCF:3a2 (3a2).4a-(3a2).3 3a2(4a-3) GCF:5 +(5).4a-(5).3 +5(4a-3) 3a2(4a-3)+5(4a-3)

(4a-3) is a common factor Factor out (4a-3)

(4a-3)(3a2+5)

Factoring quadratic trinomials where a=1

First step find the GCF

GCF:1

Find the product ac:

ax² + bx + c

x² + 5x + 6 1x6=6

look for factors of 6 whose sum is 5

6 3+2=5

Write the expression in grouping

x2+3x +2x+6

Now factor the expression by grouping

GCF:x (x).x+(x).3 x(x+3) GCF:2 +(2).x+(2).3 +2(x+3)

Now pull the GCF out and put the other numbers in bracket

x(x+3)+2(x+3) (x+3)(x+2) Method 1

Factoring quadratic trinomials where a=1

ax² + bx + c

Method 2 First step find the GCF Find the product ac: look for factors of 36 whose sum is 13

x²+13x+36 GCF:1 1x36 36 4+9=13 (x+4)(x+9)

Now take write x plus one of the factors then write x plus the other factor Note:in this method the variable should be equal to one

Factoring quadratic trinomials where a ≠ 1

First step find the GCF

ax² + bx + c

2x²+9x+10

Find the product ac:

2x10=20 20

look for factors of 20 whose sum is 9

4+5=9 2x2+4x +5x+10 GCF:2x GCF:5 (2x).x+(2x).2

Write the expression in grouping Now factor the expression by grouping Now pull the GCF out and put the other numbers in bracket

2x(x+3) +(5).x+(5).2 +5(x+3) 2x(x+3)+5(x+3) (x+3)(2x+5) GCF:1

Factoring a perfect square

a² + 2ab + b2 = (a + b)²(a + b)²

Check whether it's a perfect square by checking the square root for a and c

9x²-12x+4 First term:√9x²=3x Second term:√4=2 Middle term:2x3xx2=12x

It's a perfect square,so we must but the answer of the first term minus the answer

• f the second term in bracket and we

must but a square

(3x-2)²

Note:in perfect square the second term must be plus

Factoring a difference of two squares

Check whether it's a perfect square by checking the square root for a and c

4r²-25s² a² – b² = (a – b)(a + b) (2r3)²-(5s3)² First term:√4r6=2r3 Last term:√25s6=5s3 (2r3-5s3)(2r3+5s3)

It's a difference of two square,so we must but the answer of the first term minus and plus the answer of the last term in bracket Note:in difference of two square (a)must be minus (b) not plus

Factoring Difference of cube

a^3 – b^3= (a – b)(a^2+ ab + b^2)

check the cube root for a and c

27y3-64

It's a difference of cubes,so we must but the answer same as the expression that is up

First term:3√27y^3=3y Last term:3√64=4 (3y-4)(3y^2+3y.4+4^2) (3y-4)(9y2+12y+16)

Factoring completely

P^4-16 GCF:1

It's a difference of two squares so we need the the square root of the p^4 and 16

First term:√p^4=p² Last term:√16=4

we must but the answer of the first term minus and plus the answer of the last term in bracket

(P²)²-(4)² (P2+4)(p2-4)

Now we can factor(p²-4) because it is a difference of two squares

(P)²-(2)² (P²+4)(p+2)(p-2)