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2. Factoring is to write an expression as a product of factors. For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10. We can also do this with polynomial expressions. In this tutorial we are going to look at several ways to factor polynomial expressions. By the time I'm through with you, you will be a factoring machine. Factoring is to write an expression as a product of factors. For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10. We can also do this with polynomial expressions. In this tutorial we are going to look at several ways to factor polynomial expressions. By the time I'm through with you, you will be a factoring machine.

3. 2. Factor out the GCF of a polynomialFactor out the GCF of a polynomial 3. Factor a difference of squares.Factor a difference of squares. 4. Factor a sum or difference of cubesFactor a sum or difference of cubes. 5. Factor a trinomial of the form: Factor a trinomial of the form:. 6. Factor a trinomial of the form:Factor a trinomial of the form:. 8. Factor a polynomial with four terms by grouping.Factor a polynomial with four terms by grouping. 10. Indicate if a polynomial is a prime polynomial. Indicate if a polynomial is a prime polynomial 9. Apply the factoring strategy to factor a polynomial completely.. Apply the factoring strategy to factor a polynomial completely. 1. Find the Greatest Common Factor (GCF) of a polynomial. 7. Factor a perfect square trinomial.. Factor a perfect square trinomial. Click on the type of factoring you need help with.

4. Always look for a Greatest Common Factor FIRST!!!

5. The GCF for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. 1. Is there a number that divides into all terms? What is it? 2. Is there a variable(s) in all terms? Use the smallest exponent. 3. Divide your results into all numbers or terms. 4. Put your answer in parenthesis with the GMF in front. Always look for a Greatest Common Factor FIRST!!!

6. Example 1: Factor out the GCF:. Step 1: Identify the GCF of the polynomial. The largest monomial that we can factor out of each term is 2y. Step 2: Divide the GCF out of every term of the polynomial. Divide 2y out of every term of the poly. Step 3: Put your answer in parenthesis with the GMF in front. This process is basically the reverse of the distributive property. BBe careful. If a term of the polynomial is exactly the same as the GCF, when you divide it by the GCF you are left with 1, NOT 0!! Don’t think, “Oh I have nothing left;” there is actually a 1. As shown above when we divide 2y by 2y we get 1, so we need a 1 as the third term inside of the ( ).

7. Example 2: Factor out the GCF:. Step 1: Identify the GCF of the polynomial.GCF This time it isn't a monomial but a binomial that we have in common. Our GCF is (3x -1). Step 2: Divide the GCF out of every term of the polynomial. *Divide (3x - 1) out of both parts When we divide (3x - 1) out of the first term, we are left with x. When we divide it out of the second term, we are left with 5. That is how we get the (x + 5) for our second ( ). Step 3: Put your answer in parenthesis with the GMF in front.

8. (Must be in one of the following forms to factor with two terms)

9. 1. Are there two terms separated by a minus sign? 2. Are both terms perfect squares? What are they the square of? 3. Make two parentheses, one with a (+) and one with a (–), then fit the factor in the parentheses as below. = (4a+5)(4a-5 ) yes! yes! Note that the sum of two squares DOES NOT factor. This is the reverse of the FOIL method. Recall that factoring is the reverse of multiplication.

10. Example 1: Factor the difference of two squares:. First note that there is no GCF to factor out of this polynomial Step1: Are there two terms separated by a minus sign? Step 2: Are both terms perfect squares? What are they the square of? Step 3: Make two parentheses, one with a (+) and one with a (–), then fit the factor in the parentheses as below. yes! yes! Note that if we would multiply this out, we would get the original polynomial Example 2: Factor the difference of two squares:. Step1: Are there two terms separated by a minus sign? Step 2: Are both terms perfect squares? What are they the square of? Step 3: Make two parentheses, one with a (+) and one with a (–), then fit the factor in the parentheses as below.

11. 1. Are there two terms? cubes 2. Are both terms cubes? What are they the cubes of? 3. If there is a plus sign, use the form: yes! yes! 4. If there is a minus sign, use the form: Note that the sum (+) of t tt two cubes will factor!!

12. Step 1: Are there two terms? Step 2: Are both terms cubes? What are they the cubes of? Step 3: If there is a plus sign, use the form: yes! yes! Step 4: If there is a minus sign, use the form: Example 1: Factor the difference of two cubes: Note that this trinomial does not have a GCF.

13. (X or Pattern)

14. 1. Are there three terms? (Where the number in front of x squared is 1) (where a does not equal 1) 2. Write all factor pairs of the 1st term. 3. Write all factor pairs of the 3rd term. 4. Make two parentheses. If the third term is (+), use the sign of the 2nd term in both parentheses. If the third term is (–), use one (+) and one (–) in the parentheses. 5. Fit your factors into the parentheses so that your middle term checks out right. USE O AND I OF FOIL yes

15. Example 1: Factor the trinomial:. Step 1: Are there three terms? yes! Step 2: Write all factor pairs of the 1st term. Step 3: Write all factor pairs of the 3rd term. Step 4: Make two sets parentheses. If the third term is (+,) use the sign of the 2nd term in both parentheses. If the third term is (–), use one (+) and one (–) in the parentheses. Step 5: Fit your factors into the parentheses so that your middle term checks out right. Note that this trinomial does not have a GCF. Use O and I of the FOIL

16. Example 2: Factor the trinomial:. Step 1: Are there three terms in the parentheses? yes! Step 2: Write all factor pairs of the 1st term. Step 3: Write all factor pairs of the 3rd term. Step 4: Make two sets parentheses. If the third term is (+,) use the sign of the 2nd term in both parentheses. If the third term is (–), use one (+) and one (–) in the parentheses. Step 5: Fit your factors into the parentheses so that your middle term checks out right. We need to factor out the GCF before we tackle the trinomial part of this.GCF Use O and I of the FOIL does Note that this trinomial does have a GCF of 2y. * Factor out the GCF of 2y Don’t forget the 2y: put it in front.

17. Example 3: Factor the trinomial. Step 1: Are there three terms. yes! Step 2: Write all factor pairs of the 1st term. Step 3: Write all factor pairs of the 3rd term. Step 4: Make two parentheses. If the third term is (+,) use the sign of the 2nd term in both parentheses. If the third term is (–), use one (+) and one (–) in the parentheses. Step 5: Fit your factors into the parentheses so that your middle term checks out right. Use O and I of the FOIL Note that this trinomial does not have a GCF. This is not the answer. This is the answer.

18. Example 4: Factor the trinomial. Step 1: Are there three terms. yes! Step 2: Write all factor pairs of the 1st term. Step 3: Write all factor pairs of the 3rd term. Step 4: Make two parentheses. If the third term is (+,) use the sign of the 2nd term in both parentheses. If the third term is (–), use one (+) and one (–) in the parentheses. Step 5: Fit your factors into the parentheses so that your middle term checks out right. Note that this trinomial does not have a GCF. This is not the answer. This is the answer. Don’t give up! Use O and I of the FOIL This is not the answer.

19. 1. Are there three terms? 2. Are the first and last term squares? What are they square of? 3. Make one parenthesis using the sign from the middle term and square it. yes Step 1: Are there three terms? Example 1: Factor the trinomial: yes Step 2: Are the first and last terms squares? What are they square of? yes Step 3: Make one parenthesis using the sign from the middle term and square it.

20. (Grouping)

21. 1. Are there four terms? 2. Pair off the terms so that there is a common factor 3. Factor out a GCF from each separate binomial. 4. Write your GMF’s in one parenthesis, and the rest in another parenthesis. yes When we divide out the (x-2) out of the first term, we are left with. When we divide it out of the second term, we are left with 5. That is how we get the for our second ( ).

22. Step 1: Are there four terms? Step 2: Pair off the terms so that there is a common factor Step 3: Factor out a GCF from each separate binomial. Step 4: Write your GMF’s in one parenthesis, and the rest in another parenthesis. yes When we divide out the (x+3) out of the first term, we are left with When we divide it out of the second term, we are left with 2. That is how we get the for our second ( ). Example 1: Factor by grouping:

23. . (Check to see if any of your final answers will factor further )

24.  I. GCF: Always check for the GCF first, no matter what  II. Binomials: Are there two terms?  III. Trinomials: Are there three terms? a. b. Reverse of the FOIL method: c. Perfect square trinomial :  IV. Polynomials with four terms: Factor by grouping

25. Step1: Always check for the GCF first, no matter what Example 1: Factor completely: *Factor a 3 out of every term Step 2: There are three terms, and the first and last terms are perfect squares. Step 3: Make one parenthesis using the sign from the middle term and square it. Step 4: Don’t forget to put the GMF in front of the factored problem.

26. Example 2: Factor completely: Step1: Always check for the GCF first, no matter what. But there is not one. Step 2: There are two terms, with a minus sign, and both terms are squares. Step 3 : Look at the 2 nd factor; there are two terms, with a minus sign, and both terms are squares. Factor that term, also. Now you are done.

27. . (If nothing can be done to the original expression, then it is PRIME)

28.   Not every polynomial is factorable. Just like not every number has a factor other than 1 or itself. A prime number is a number that has exactly two factors, 1 and itself. 2, 3, and 5 are examples of prime numbers.   The same thing can occur with polynomials. If a polynomial is not factorable we say that it is a prime polynomial.   Sometimes you will not know it is prime until you start looking for factors of it. Once you have exhausted all possibilities, then you can call it prime. Be careful. Do not think because you could not factor it on the first try that it is prime. You must go through ALL possibilities first before declaring it prime.

29. 1. Are there three terms? 2. Write all factor pairs of the 1st term. 3. Write all factor pairs of the 3rd term. 4. Make two sets of parentheses. If the third term is (+,) use the sign of the 2nd term in both parentheses. If the third term is (–), use one (+) and one (–) in the parentheses. 5. Fit your factors into the parenthesis so that your middle term checks out right. USE O AND I OF FOIL yes This polynomial is prime. Don’t give up!

30. 1. Are there two terms? cubes 2. Are both terms squares or cubes? yes no So we can not use difference of squares or difference or sum of cubes. This polynomial is prime. Don’t give up!