In the previous section we were multiplying a monomial by a polynomial expression like this… 3𝑎 𝑎 2 −5𝑎+4 =3 𝑎 3 −15 𝑎 2 +12𝑎 FACTORING

In the previous section we were multiplying a monomial by a polynomial expression like this… 3𝑎 𝑎 2 −5𝑎+4 =3 𝑎 3 −15 𝑎 2 +12𝑎 FACTORING In this section we are going to go backwards. What expressions were used to create a polynomial expression through multiplication. This process is called factoring.

In the previous section we were multiplying a monomial by a polynomial expression like this… 3𝑎 𝑎 2 −5𝑎+4 =3 𝑎 3 −15 𝑎 2 +12𝑎 FACTORING In this section we are going to go backwards. What expressions were used to create a polynomial expression through multiplication. This process is called factoring. There are some different ways to factor which depend on the expression your faced with. We’ll start with what we call “common terms”.

COMMON TERMS FACTORING Look at the given expression : 2 𝑥 2 +8𝑥+12

FACTORING COMMON TERMS Look at the given expression : 2 𝑥 2 +8𝑥+12 Some expressions will only have one common term. It could be an integer or variable. As you can see, all three terms are divisible by 2. The “common term” is extracted and the result will be a monomial outside brackets and a polynomial inside with each term divided by the common term .

FACTORING COMMON TERMS Look at the given expression : 2 𝑥 2 +8𝑥+12 Some expressions will only have one common term. It could be an integer or variable. As you can see, all three terms are divisible by 2. The “common term” is extracted and the result will be a monomial outside brackets and a polynomial inside with each term divided by the common term . Like this… 2 2 𝑥 2 2 + 8𝑥 2 + 12 2

FACTORING COMMON TERMS Look at the given expression : 2 𝑥 2 +8𝑥+12 Some expressions will only have one common term. It could be an integer or variable. As you can see, all three terms are divisible by 2. The “common term” is extracted and the result will be a monomial outside brackets and a polynomial inside with each term divided by the common term . Like this… 2 2 𝑥 2 2 + 8𝑥 2 + 12 2 =2 𝑥 2 +4𝑥+6

FACTORING COMMON TERMS Let’s try another that has multiple terms as common… −3 𝑎 4 𝑏 2 +9𝑎 𝑏 5

FACTORING COMMON TERMS Let’s try another that has multiple terms as common… −3 𝑎 4 𝑏 2 +9𝑎 𝑏 5 common term is 3𝑎 𝑏 2 ( IT’S THE GCF)

FACTORING COMMON TERMS Let’s try another that has multiple terms as common… −3 𝑎 4 𝑏 2 +9𝑎 𝑏 5 common term is 3𝑎 𝑏 2 ( IT’S THE GCF) =3𝑎 𝑏 2 −3 𝑎 4 𝑏 2 3𝑎 𝑏 2 + 9𝑎 𝑏 5 3𝑎 𝑏 2

FACTORING =3𝑎 𝑏 2 (− 𝑎 3 +3 𝑏 3 ) COMMON TERMS Let’s try another that has multiple terms as common… −3 𝑎 4 𝑏 2 +9𝑎 𝑏 5 common term is 3𝑎 𝑏 2 ( IT’S THE GCF) =3𝑎 𝑏 2 −3 𝑎 4 𝑏 2 3𝑎 𝑏 2 + 9𝑎 𝑏 5 3𝑎 𝑏 2 =3𝑎 𝑏 2 (− 𝑎 3 +3 𝑏 3 )

COMMON TERMS FACTORING Let’s try another that has multiple terms as common… −3 𝑎 4 𝑏 2 +9𝑎 𝑏 5 common term is 3𝑎 𝑏 2 ( IT’S THE GCF) =3𝑎 𝑏 2 −3 𝑎 4 𝑏 2 3𝑎 𝑏 2 + 9𝑎 𝑏 5 3𝑎 𝑏 2 =3𝑎 𝑏 2 (− 𝑎 3 +3 𝑏 3 ) ** usually if a negative appears in front, you will take the a negative out as well…

COMMON TERMS FACTORING Let’s try another that has multiple terms as common… −3 𝑎 4 𝑏 2 +9𝑎 𝑏 5 common term is 3𝑎 𝑏 2 ( IT’S THE GCF) =3𝑎 𝑏 2 −3 𝑎 4 𝑏 2 3𝑎 𝑏 2 + 9𝑎 𝑏 5 3𝑎 𝑏 2 =3𝑎 𝑏 2 (− 𝑎 3 +3 𝑏 3 ) ** usually if a negative appears in front, you will take the a negative out as well… =−3𝑎 𝑏 2 −3 𝑎 4 𝑏 2 −3𝑎 𝑏 2 + 9𝑎 𝑏 5 −3𝑎 𝑏 2 =−3𝑎 𝑏 2 ( 𝑎 3 −3 𝑏 3 )

FACTORING COMMON TERMS There is another common term factoring method that uses binomials as common factors. Also called “factoring by grouping”

FACTORING COMMON TERMS There is another common term factoring method that uses binomials as common factors. Also called “factoring by grouping” Look at this expression … 2x 𝑥+4 −5(𝑥+4)

FACTORING COMMON TERMS There is another common term factoring method that uses binomials as common factors. Also called “factoring by grouping” Look at this expression … 2x 𝑥+4 −5(𝑥+4) - notice how (𝑥+4) is a common binomial for each term

FACTORING COMMON TERMS There is another common term factoring method that uses binomials as common factors. Also called “factoring by grouping” Look at this expression … 2x 𝑥+4 −5(𝑥+4) - notice how (𝑥+4) is a common binomial for each term - I like to call these common binomials “twins”

FACTORING COMMON TERMS There is another common term factoring method that uses binomials as common factors. Also called “factoring by grouping” Look at this expression … 2x 𝑥+4 −5(𝑥+4) - notice how (𝑥+4) is a common binomial for each term - I like to call these common binomials “twins” Your answer becomes (2𝑥−5)(𝑥+4) You simply have your “twin” as one of the binomial terms, and whatever appears outside as the other binomial term.

FACTORING COMMON TERMS There is another common term factoring method that uses binomials as common factors. Also called “factoring by grouping” Look at this expression … 2x 𝑥+4 −5(𝑥+4) - notice how (𝑥+4) is a common binomial for each term - I like to call these common binomials “twins” Your answer becomes (2𝑥−5)(𝑥+4) You simply have your “twin” as one of the binomial terms, and whatever appears outside as the other binomial term. So for example : 3𝑎 𝑥−𝑦 +5𝑏(𝑥−𝑦) becomes (3𝑎+5𝑏)(𝑥−𝑦)

FACTORING COMMON TERMS We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor 𝑎𝑥+2𝑏𝑥−2𝑎𝑦−4𝑏𝑦

FACTORING COMMON TERMS We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor 𝑎𝑥+2𝑏𝑥−2𝑎𝑦−4𝑏𝑦 STEPS : 1. Find common terms between pairs of terms

FACTORING COMMON TERMS We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor 𝑎𝑥+2𝑏𝑥−2𝑎𝑦−4𝑏𝑦 𝑥 𝑎𝑥 𝑥 + 2𝑏𝑥 𝑥 −2𝑦 −2𝑎𝑦 −2𝑦 + −4𝑏𝑦 −2𝑦 STEPS : 1. Find common terms between pairs of terms - 𝑎𝑥+2𝑏𝑥 share 𝑥 , −2𝑎𝑦−4𝑏𝑦 share −2𝑦

FACTORING COMMON TERMS We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor 𝑎𝑥+2𝑏𝑥−2𝑎𝑦−4𝑏𝑦 𝑥 𝑎𝑥 𝑥 + 2𝑏𝑥 𝑥 −2𝑦 −2𝑎𝑦 −2𝑦 + −4𝑏𝑦 −2𝑦 =𝑥 𝑎+2𝑏 −2𝑦(𝑎+2𝑏) STEPS : 1. Find common terms between pairs of terms - 𝑎𝑥+2𝑏𝑥 share 𝑥 , −2𝑎𝑦−4𝑏𝑦 share −2𝑦 2. Extract your common term

FACTORING COMMON TERMS We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor 𝑎𝑥+2𝑏𝑥−2𝑎𝑦−4𝑏𝑦 𝑥 𝑎𝑥 𝑥 + 2𝑏𝑥 𝑥 −2𝑦 −2𝑎𝑦 −2𝑦 + −4𝑏𝑦 −2𝑦 =𝑥 𝑎+2𝑏 −2𝑦(𝑎+2𝑏) (𝑥−2𝑦)(𝑎+2𝑏) STEPS : 1. Find common terms between pairs of terms - 𝑎𝑥+2𝑏𝑥 share 𝑥 , −2𝑎𝑦−4𝑏𝑦 share −2𝑦 2. Extract your common term 3. Write your answer using your twins and common terms

FACTORING COMMON TERMS We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor 𝑎𝑥+2𝑏𝑥−2𝑎𝑦−4𝑏𝑦 𝑥 𝑎𝑥 𝑥 + 2𝑏𝑥 𝑥 −2𝑦 −2𝑎𝑦 −2𝑦 + −4𝑏𝑦 −2𝑦 =𝑥 𝑎+2𝑏 −2𝑦(𝑎+2𝑏) (𝑥−2𝑦)(𝑎+2𝑏) STEPS : 1. Find common terms between pairs of terms - 𝑎𝑥+2𝑏𝑥 share 𝑥 , −2𝑎𝑦−4𝑏𝑦 share −2𝑦 2. Extract your common term 3. Write your answer using your twins and common terms ** if you don’t get twins in step 2, try rewriting the order of the terms

COMMON TERMS FACTORING EXAMPLE : Factor 2m−3𝑛+𝑛𝑚−6

FACTORING COMMON TERMS EXAMPLE : Factor 2m−3𝑛+𝑛𝑚−6 STEPS : 1. Find common terms between pairs of terms

FACTORING COMMON TERMS EXAMPLE : Factor 2m−3𝑛+𝑛𝑚−6 =2𝑚+𝑛𝑚−3𝑛−6 STEPS : 1. Find common terms between pairs of terms. ** there isn’t any common terms between pairs as written so rewrite the order of terms

FACTORING COMMON TERMS EXAMPLE : Factor 2m−3𝑛+𝑛𝑚−6 =2𝑚+𝑛𝑚−3𝑛−6 STEPS : 1. Find common terms between pairs of terms. ** there isn’t any common terms between pairs as written so rewrite the order of terms - now find the common terms. I like to place a dividing line in between the pairs

FACTORING COMMON TERMS EXAMPLE : Factor 2m−3𝑛+𝑛𝑚−6 =2𝑚+𝑛𝑚−3𝑛−6 =𝑚 2𝑚 𝑚 + 𝑛𝑚 𝑚 −3 −3𝑛 −3 + −6 −3 𝑚 2+𝑛 −3(𝑛+2) STEPS : 1. Find common terms between pairs of terms. ** there isn’t any common terms between pairs as written so rewrite the order of terms - now find the common terms. I like to place a dividing line in between the pairs 2. Extract the common terms

FACTORING COMMON TERMS EXAMPLE : Factor 2m−3𝑛+𝑛𝑚−6 =2𝑚+𝑛𝑚−3𝑛−6 =𝑚 2𝑚 𝑚 + 𝑛𝑚 𝑚 −3 −3𝑛 −3 + −6 −3 𝑚 2+𝑛 −3(𝑛+2) (𝑚−3)(𝑛+2) STEPS : 1. Find common terms between pairs of terms. ** there isn’t any common terms between pairs as written so rewrite the order of terms - now find the common terms. I like to place a dividing line in between the pairs 2. Extract the common terms 3. Get your answer ** yes, 𝑛+2 =(2+𝑛)

FACTORING COMMON TERMS One more EXAMPLE :

FACTORING COMMON TERMS One more EXAMPLE : =2 𝑥 2 12 𝑥 3 2 𝑥 2 + 10 𝑥 2 2 𝑥 2 −5 −30𝑥 −5 + −25 −5 - extract your common term

FACTORING COMMON TERMS One more EXAMPLE : =2 𝑥 2 12 𝑥 3 2 𝑥 2 + 10 𝑥 2 2 𝑥 2 −5 −30𝑥 −5 + −25 −5 =2 𝑥 2 6𝑥+5 −5(6𝑥+5) =(2 𝑥 2 −5)(6𝑥+5) - look for twins and get your answer

FACTORING COMMON TERMS The next type of expression we are going to factor are called difference of squares. They are always binomials where terms are separated by a negative sign and the integers are perfect squares and exponents are even.

FACTORING COMMON TERMS The next type of expression we are going to factor are called difference of squares. They are always binomials where terms are separated by a negative sign and the integers are perfect squares and exponents are even. Some examples : 𝑥 2 −9 , 4 𝑎 2 −25 , 81 𝑥 2 −49 𝑦 2

FACTORING COMMON TERMS The next type of expression we are going to factor are called difference of squares. They are always binomials where terms are separated by a negative sign and the integers are perfect squares and exponents are even. Some examples : 𝑥 2 −9 , 4 𝑎 2 −25 , 81 𝑥 2 −49 𝑦 2 IN General : 𝑎 2 − 𝑏 2 =(𝑎+𝑏)(𝑎−𝑏) - these types factor into square roots of the terms into two binomials with one positive and one negative sign.

FACTORING COMMON TERMS The next type of expression we are going to factor are called difference of squares. They are always binomials where terms are separated by a negative sign and the integers are perfect squares and exponents are even. Some examples : 𝑥 2 −9 , 4 𝑎 2 −25 , 81 𝑥 2 −49 𝑦 2 IN General : 𝑎 2 − 𝑏 2 =(𝑎+𝑏)(𝑎−𝑏) - these types factor into square roots of the terms into two binomials with one positive and one negative sign. You should know your square roots…and the square root of an exponent is just dividing that exponent by 2. EXAMPLE : 𝑥 2 = 𝑥 𝑥 4 = 𝑥 2 𝑥 6 = 𝑥 3 and so on….

COMMON TERMS FACTORING EXAMPLE : Factor 𝑥 2 −4

FACTORING COMMON TERMS EXAMPLE : Factor 𝑥 2 −4 𝑥 2 =𝑥 4 =2 𝑥 2 =𝑥 4 =2 STEPS : 1. Find the square roots of each term.

FACTORING COMMON TERMS EXAMPLE : Factor 𝑥 2 −4 𝑥 2 =𝑥 4 =2 =(𝑥+2)(𝑥−2) 𝑥 2 =𝑥 4 =2 =(𝑥+2)(𝑥−2) STEPS : 1. Find the square roots of each term. 2. Put in the form (𝑎+𝑏)(𝑎−𝑏)

FACTORING COMMON TERMS EXAMPLE : Factor 9𝑥 2 −16 STEPS : 1. Find the square roots of each term.

FACTORING COMMON TERMS EXAMPLE : Factor 9𝑥 2 −16 9𝑥 2 =3𝑥 16 =4 9𝑥 2 =3𝑥 16 =4 STEPS : 1. Find the square roots of each term.

FACTORING COMMON TERMS EXAMPLE : Factor 9𝑥 2 −16 9𝑥 2 =3𝑥 16 =4 9𝑥 2 =3𝑥 16 =4 =(3𝑥+4)(3𝑥−4) STEPS : 1. Find the square roots of each term. 2. Put in the form (𝑎+𝑏)(𝑎−𝑏)

FACTORING COMMON TERMS Some problems will require you combine methods...

FACTORING COMMON TERMS EXAMPLE : Factor 2𝑥 2 −50 𝑦 2 STEPS : 1. Find common term

FACTORING COMMON TERMS EXAMPLE : Factor 2𝑥 2 −50 𝑦 2 =2 2 𝑥 2 2 − 50 𝑦 2 2 STEPS : 1. Find common term

FACTORING COMMON TERMS EXAMPLE : Factor 2𝑥 2 −50 𝑦 2 =2 2 𝑥 2 2 − 50 𝑦 2 2 =2( 𝑥 2 −25 𝑦 2 ) STEPS : 1. Find common term

FACTORING COMMON TERMS EXAMPLE : Factor 2𝑥 2 −50 𝑦 2 =2 2 𝑥 2 2 − 50 𝑦 2 2 =2( 𝑥 2 −25 𝑦 2 ) 𝑥 2 =𝑥 25𝑦 2 =5𝑦 STEPS : 1. Find common term 2. Factor difference of squares

FACTORING COMMON TERMS EXAMPLE : Factor 2𝑥 2 −50 𝑦 2 =2 2 𝑥 2 2 − 50 𝑦 2 2 =2( 𝑥 2 −25 𝑦 2 ) 𝑥 2 =𝑥 25𝑦 2 =5𝑦 =2(𝑥+5𝑦)(𝑥−5𝑦) STEPS : 1. Find common term 2. Factor difference of squares ** notice how the common term remained in the answer