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In the previous section we were multiplying a monomial by a polynomial expression like thisβ¦
3π π 2 β5π+4 =3 π 3 β15 π 2 +12π FACTORING
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In the previous section we were multiplying a monomial by a polynomial expression like thisβ¦
3π π 2 β5π+4 =3 π 3 β15 π 2 +12π 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.
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In the previous section we were multiplying a monomial by a polynomial expression like thisβ¦
3π π 2 β5π+4 =3 π 3 β15 π 2 +12π 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. Weβll start with what we call βcommon termsβ.
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COMMON TERMS FACTORING Look at the given expression : 2 π₯ 2 +8π₯+12
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FACTORING COMMON TERMS Look at the given expression : 2 π₯ 2 +8π₯+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 βcommon termβ is extracted and the result will be a monomial outside brackets and a polynomial inside with each term divided by the common term .
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FACTORING COMMON TERMS Look at the given expression : 2 π₯ 2 +8π₯+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 βcommon termβ is extracted and the result will be a monomial outside brackets and a polynomial inside with each term divided by the common term . Like thisβ¦ 2 2 π₯ π₯
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FACTORING COMMON TERMS Look at the given expression : 2 π₯ 2 +8π₯+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 βcommon termβ is extracted and the result will be a monomial outside brackets and a polynomial inside with each term divided by the common term . Like thisβ¦ 2 2 π₯ π₯ =2 π₯ 2 +4π₯+6
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FACTORING COMMON TERMS
Letβs try another that has multiple terms as commonβ¦ β3 π 4 π 2 +9π π 5
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FACTORING COMMON TERMS
Letβs try another that has multiple terms as commonβ¦ β3 π 4 π 2 +9π π 5 common term is 3π π 2 ( ITβS THE GCF)
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FACTORING COMMON TERMS
Letβs try another that has multiple terms as commonβ¦ β3 π 4 π 2 +9π π 5 common term is 3π π 2 ( ITβS THE GCF) =3π π 2 β3 π 4 π 2 3π π π π 5 3π π 2
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FACTORING =3π π 2 (β π 3 +3 π 3 ) COMMON TERMS
Letβs try another that has multiple terms as commonβ¦ β3 π 4 π 2 +9π π 5 common term is 3π π 2 ( ITβS THE GCF) =3π π 2 β3 π 4 π 2 3π π π π 5 3π π 2 =3π π 2 (β π 3 +3 π 3 )
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COMMON TERMS FACTORING Letβs try another that has multiple terms as commonβ¦ β3 π 4 π 2 +9π π 5 common term is 3π π 2 ( ITβS THE GCF) =3π π 2 β3 π 4 π 2 3π π π π 5 3π π 2 =3π π 2 (β π 3 +3 π 3 ) ** usually if a negative appears in front, you will take the a negative out as wellβ¦
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COMMON TERMS FACTORING Letβs try another that has multiple terms as commonβ¦ β3 π 4 π 2 +9π π 5 common term is 3π π 2 ( ITβS THE GCF) =3π π 2 β3 π 4 π 2 3π π π π 5 3π π 2 =3π π 2 (β π 3 +3 π 3 ) ** usually if a negative appears in front, you will take the a negative out as wellβ¦ =β3π π 2 β3 π 4 π 2 β3π π π π 5 β3π π 2 =β3π π 2 ( π 3 β3 π 3 )
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FACTORING COMMON TERMS
There is another common term factoring method that uses binomials as common factors. Also called βfactoring by groupingβ
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FACTORING COMMON TERMS
There is another common term factoring method that uses binomials as common factors. Also called βfactoring by groupingβ Look at this expression β¦ 2x π₯+4 β5(π₯+4)
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FACTORING COMMON TERMS
There is another common term factoring method that uses binomials as common factors. Also called βfactoring by groupingβ Look at this expression β¦ 2x π₯+4 β5(π₯+4) - notice how (π₯+4) is a common binomial for each term
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FACTORING COMMON TERMS
There is another common term factoring method that uses binomials as common factors. Also called βfactoring by groupingβ Look at this expression β¦ 2x π₯+4 β5(π₯+4) - notice how (π₯+4) is a common binomial for each term - I like to call these common binomials βtwinsβ
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FACTORING COMMON TERMS
There is another common term factoring method that uses binomials as common factors. Also called βfactoring by groupingβ Look at this expression β¦ 2x π₯+4 β5(π₯+4) - notice how (π₯+4) is a common binomial for each term - I like to call these common binomials βtwinsβ Your answer becomes (2π₯β5)(π₯+4) You simply have your βtwinβ as one of the binomial terms, and whatever appears outside as the other binomial term.
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FACTORING COMMON TERMS
There is another common term factoring method that uses binomials as common factors. Also called βfactoring by groupingβ Look at this expression β¦ 2x π₯+4 β5(π₯+4) - notice how (π₯+4) is a common binomial for each term - I like to call these common binomials βtwinsβ Your answer becomes (2π₯β5)(π₯+4) You simply have your βtwinβ as one of the binomial terms, and whatever appears outside as the other binomial term. So for example : π π₯βπ¦ +5π(π₯βπ¦) becomes (3π+5π)(π₯βπ¦)
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FACTORING COMMON TERMS
We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor ππ₯+2ππ₯β2ππ¦β4ππ¦
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FACTORING COMMON TERMS
We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor ππ₯+2ππ₯β2ππ¦β4ππ¦ STEPS : Find common terms between pairs of terms
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FACTORING COMMON TERMS
We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor ππ₯+2ππ₯β2ππ¦β4ππ¦ π₯ ππ₯ π₯ + 2ππ₯ π₯ β2π¦ β2ππ¦ β2π¦ + β4ππ¦ β2π¦ STEPS : Find common terms between pairs of terms - ππ₯+2ππ₯ share π₯ , β2ππ¦β4ππ¦ share β2π¦
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FACTORING COMMON TERMS
We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor ππ₯+2ππ₯β2ππ¦β4ππ¦ π₯ ππ₯ π₯ + 2ππ₯ π₯ β2π¦ β2ππ¦ β2π¦ + β4ππ¦ β2π¦ =π₯ π+2π β2π¦(π+2π) STEPS : Find common terms between pairs of terms - ππ₯+2ππ₯ share π₯ , β2ππ¦β4ππ¦ share β2π¦ 2. Extract your common term
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FACTORING COMMON TERMS
We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor ππ₯+2ππ₯β2ππ¦β4ππ¦ π₯ ππ₯ π₯ + 2ππ₯ π₯ β2π¦ β2ππ¦ β2π¦ + β4ππ¦ β2π¦ =π₯ π+2π β2π¦(π+2π) (π₯β2π¦)(π+2π) STEPS : Find common terms between pairs of terms - ππ₯+2ππ₯ share π₯ , β2ππ¦β4ππ¦ share β2π¦ 2. Extract your common term 3. Write your answer using your twins and common terms
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FACTORING COMMON TERMS
We will use this method for two types of problems. The first type are four term polynomials. EXAMPLE : Factor ππ₯+2ππ₯β2ππ¦β4ππ¦ π₯ ππ₯ π₯ + 2ππ₯ π₯ β2π¦ β2ππ¦ β2π¦ + β4ππ¦ β2π¦ =π₯ π+2π β2π¦(π+2π) (π₯β2π¦)(π+2π) STEPS : Find common terms between pairs of terms - ππ₯+2ππ₯ share π₯ , β2ππ¦β4ππ¦ share β2π¦ 2. Extract your common term 3. Write your answer using your twins and common terms ** if you donβt get twins in step 2, try rewriting the order of the terms
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COMMON TERMS FACTORING EXAMPLE : Factor 2mβ3π+ππβ6
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FACTORING COMMON TERMS EXAMPLE : Factor 2mβ3π+ππβ6
STEPS : Find common terms between pairs of terms
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FACTORING COMMON TERMS EXAMPLE : Factor 2mβ3π+ππβ6 =2π+ππβ3πβ6
STEPS : Find common terms between pairs of terms. ** there isnβt any common terms between pairs as written so rewrite the order of terms
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FACTORING COMMON TERMS EXAMPLE : Factor 2mβ3π+ππβ6 =2π+ππβ3πβ6
STEPS : Find common terms between pairs of terms. ** there isnβt 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
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FACTORING COMMON TERMS EXAMPLE : Factor 2mβ3π+ππβ6 =2π+ππβ3πβ6
=π 2π π + ππ π β3 β3π β3 + β6 β3 π 2+π β3(π+2) STEPS : Find common terms between pairs of terms. ** there isnβt 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
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FACTORING COMMON TERMS EXAMPLE : Factor 2mβ3π+ππβ6 =2π+ππβ3πβ6
=π 2π π + ππ π β3 β3π β3 + β6 β3 π 2+π β3(π+2) (πβ3)(π+2) STEPS : Find common terms between pairs of terms. ** there isnβt 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, π+2 =(2+π)
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FACTORING COMMON TERMS One more EXAMPLE :
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FACTORING COMMON TERMS One more EXAMPLE :
=2 π₯ π₯ 3 2 π₯ π₯ 2 2 π₯ 2 β5 β30π₯ β5 + β25 β5 - extract your common term
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FACTORING COMMON TERMS One more EXAMPLE :
=2 π₯ π₯ 3 2 π₯ π₯ 2 2 π₯ 2 β5 β30π₯ β5 + β25 β5 =2 π₯ 2 6π₯+5 β5(6π₯+5) =(2 π₯ 2 β5)(6π₯+5) - look for twins and get your answer
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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.
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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 : π₯ 2 β9 , 4 π 2 β25 , 81 π₯ 2 β49 π¦ 2
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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 : π₯ 2 β9 , 4 π 2 β25 , 81 π₯ 2 β49 π¦ 2 IN General : π 2 β π 2 =(π+π)(πβπ) - these types factor into square roots of the terms into two binomials with one positive and one negative sign.
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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 : π₯ 2 β9 , 4 π 2 β25 , 81 π₯ 2 β49 π¦ 2 IN General : π 2 β π 2 =(π+π)(πβπ) - these types factor into square roots of the terms into two binomials with one positive and one negative sign. You should know your square rootsβ¦and the square root of an exponent is just dividing that exponent by 2. EXAMPLE : π₯ 2 = π₯ π₯ 4 = π₯ 2 π₯ 6 = π₯ 3 and so onβ¦.
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COMMON TERMS FACTORING EXAMPLE : Factor π₯ 2 β4
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FACTORING COMMON TERMS EXAMPLE : Factor π₯ 2 β4 π₯ 2 =π₯ 4 =2
π₯ 2 =π₯ 4 =2 STEPS : Find the square roots of each term.
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FACTORING COMMON TERMS EXAMPLE : Factor π₯ 2 β4 π₯ 2 =π₯ 4 =2 =(π₯+2)(π₯β2)
π₯ 2 =π₯ 4 =2 =(π₯+2)(π₯β2) STEPS : Find the square roots of each term. 2. Put in the form (π+π)(πβπ)
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FACTORING COMMON TERMS EXAMPLE : Factor 9π₯ 2 β16
STEPS : Find the square roots of each term.
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FACTORING COMMON TERMS EXAMPLE : Factor 9π₯ 2 β16 9π₯ 2 =3π₯ 16 =4
9π₯ 2 =3π₯ 16 =4 STEPS : Find the square roots of each term.
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FACTORING COMMON TERMS EXAMPLE : Factor 9π₯ 2 β16 9π₯ 2 =3π₯ 16 =4
9π₯ 2 =3π₯ 16 =4 =(3π₯+4)(3π₯β4) STEPS : Find the square roots of each term. 2. Put in the form (π+π)(πβπ)
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FACTORING COMMON TERMS
Some problems will require you combine methods...
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FACTORING COMMON TERMS EXAMPLE : Factor 2π₯ 2 β50 π¦ 2
STEPS : Find common term
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FACTORING COMMON TERMS EXAMPLE : Factor 2π₯ 2 β50 π¦ 2
=2 2 π₯ 2 2 β 50 π¦ 2 2 STEPS : Find common term
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FACTORING COMMON TERMS EXAMPLE : Factor 2π₯ 2 β50 π¦ 2
=2 2 π₯ 2 2 β 50 π¦ 2 2 =2( π₯ 2 β25 π¦ 2 ) STEPS : Find common term
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FACTORING COMMON TERMS EXAMPLE : Factor 2π₯ 2 β50 π¦ 2
=2 2 π₯ 2 2 β 50 π¦ 2 2 =2( π₯ 2 β25 π¦ 2 ) π₯ 2 =π₯ 25π¦ 2 =5π¦ STEPS : Find common term 2. Factor difference of squares
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FACTORING COMMON TERMS EXAMPLE : Factor 2π₯ 2 β50 π¦ 2
=2 2 π₯ 2 2 β 50 π¦ 2 2 =2( π₯ 2 β25 π¦ 2 ) π₯ 2 =π₯ 25π¦ 2 =5π¦ =2(π₯+5π¦)(π₯β5π¦) STEPS : Find common term 2. Factor difference of squares ** notice how the common term remained in the answer
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