SOLUTION: 64. Factor each polynomial. h^2-9hs+9s^2 26. Factor using AC method 21x^2+2x-3 72.Factor 3x^2-18x-48 18. Factor 2h^2-h-3

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 64. Factor each polynomial. h^2-9hs+9s^2 26. Factor using AC method 21x^2+2x-3 72.Factor 3x^2-18x-48 18. Factor 2h^2-h-3       Log On


   

Question 161761: 64.
Factor each polynomial.
h^2-9hs+9s^2
26.
Factor using AC method
21x^2+2x-3
72.Factor
3x^2-18x-48
18. Factor
2h^2-h-3
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Factor each polynomial.
h%5E2-9hs%2B9s%5E2+
Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 9hs
First Number---- Second Number---- Sum
1………………………...........9………………………….........10
3………………………...........3………………………….........6
-1……………………..........-9………………………........…-10
-3……………………..........-3………………………........…-6
None of these factors add to 9hs.
polynomial h%5E2-9hs%2B9s%5E2+ cannot be factored

26.
Factor using AC method
21x%5E2%2B2x-3
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 21x%5E2%2B2x-3, we can see that the first coefficient is 21, the second coefficient is 2, and the last term is -3.



Now multiply the first coefficient 21 by the last term -3 to get %2821%29%28-3%29=-63.



Now the question is: what two whole numbers multiply to -63 (the previous product) and add to the second coefficient 2?



To find these two numbers, we need to list all of the factors of -63 (the previous product).



Factors of -63:

1,3,7,9,21,63

-1,-3,-7,-9,-21,-63



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to -63.

1*(-63) = -63
3*(-21) = -63
7*(-9) = -63
(-1)*(63) = -63
(-3)*(21) = -63
(-7)*(9) = -63


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:



First NumberSecond NumberSum
1-631+(-63)=-62
3-213+(-21)=-18
7-97+(-9)=-2
-163-1+63=62
-321-3+21=18
-79-7+9=2




From the table, we can see that the two numbers -7 and 9 add to 2 (the middle coefficient).



So the two numbers -7 and 9 both multiply to -63 and add to 2



Now replace the middle term 2x with -7x%2B9x. Remember, -7 and 9 add to 2. So this shows us that -7x%2B9x=2x.



21x%5E2%2Bhighlight%28-7x%2B9x%29-3 Replace the second term 2x with -7x%2B9x.



%2821x%5E2-7x%29%2B%289x-3%29 Group the terms into two pairs.



7x%283x-1%29%2B%289x-3%29 Factor out the GCF 7x from the first group.



7x%283x-1%29%2B3%283x-1%29 Factor out 3 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%287x%2B3%29%283x-1%29 Combine like terms. Or factor out the common term 3x-1



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Answer:



So 21%2Ax%5E2%2B2%2Ax-3 factors to %287x%2B3%29%283x-1%29.



In other words, 21%2Ax%5E2%2B2%2Ax-3=%287x%2B3%29%283x-1%29.



Note: you can check the answer by expanding %287x%2B3%29%283x-1%29 to get 21%2Ax%5E2%2B2%2Ax-3 or by graphing the original expression and the answer (the two graphs should be identical).



72.Factor
3x%5E2-18x-48+
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


3%2Ax%5E2-18%2Ax-48 Start with the given expression.



3%28x%5E2-6x-16%29 Factor out the GCF 3.



Now let's try to factor the inner expression x%5E2-6x-16



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Looking at the expression x%5E2-6x-16, we can see that the first coefficient is 1, the second coefficient is -6, and the last term is -16.



Now multiply the first coefficient 1 by the last term -16 to get %281%29%28-16%29=-16.



Now the question is: what two whole numbers multiply to -16 (the previous product) and add to the second coefficient -6?



To find these two numbers, we need to list all of the factors of -16 (the previous product).



Factors of -16:

1,2,4,8,16

-1,-2,-4,-8,-16



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to -16.

1*(-16) = -16
2*(-8) = -16
4*(-4) = -16
(-1)*(16) = -16
(-2)*(8) = -16
(-4)*(4) = -16


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -6:



First NumberSecond NumberSum
1-161+(-16)=-15
2-82+(-8)=-6
4-44+(-4)=0
-116-1+16=15
-28-2+8=6
-44-4+4=0




From the table, we can see that the two numbers 2 and -8 add to -6 (the middle coefficient).



So the two numbers 2 and -8 both multiply to -16 and add to -6



Now replace the middle term -6x with 2x-8x. Remember, 2 and -8 add to -6. So this shows us that 2x-8x=-6x.



x%5E2%2Bhighlight%282x-8x%29-16 Replace the second term -6x with 2x-8x.



%28x%5E2%2B2x%29%2B%28-8x-16%29 Group the terms into two pairs.



x%28x%2B2%29%2B%28-8x-16%29 Factor out the GCF x from the first group.



x%28x%2B2%29-8%28x%2B2%29 Factor out 8 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28x-8%29%28x%2B2%29 Combine like terms. Or factor out the common term x%2B2



--------------------------------------------------



So 3%28x%5E2-6x-16%29 then factors further to 3%28x-8%29%28x%2B2%29



===============================================================



Answer:



So 3%2Ax%5E2-18%2Ax-48 completely factors to 3%28x-8%29%28x%2B2%29.



In other words, 3%2Ax%5E2-18%2Ax-48=3%28x-8%29%28x%2B2%29.



Note: you can check the answer by expanding 3%28x-8%29%28x%2B2%29 to get 3%2Ax%5E2-18%2Ax-48 or by graphing the original expression and the answer (the two graphs should be identical).




18. Factor
2h%5E2-h-3
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 2x%5E2-x-3, we can see that the first coefficient is 2, the second coefficient is -1, and the last term is -3.



Now multiply the first coefficient 2 by the last term -3 to get %282%29%28-3%29=-6.



Now the question is: what two whole numbers multiply to -6 (the previous product) and add to the second coefficient -1?



To find these two numbers, we need to list all of the factors of -6 (the previous product).



Factors of -6:

1,2,3,6

-1,-2,-3,-6



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to -6.

1*(-6) = -6
2*(-3) = -6
(-1)*(6) = -6
(-2)*(3) = -6


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -1:



First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1




From the table, we can see that the two numbers 2 and -3 add to -1 (the middle coefficient).



So the two numbers 2 and -3 both multiply to -6 and add to -1



Now replace the middle term -1x with 2x-3x. Remember, 2 and -3 add to -1. So this shows us that 2x-3x=-1x.



2x%5E2%2Bhighlight%282x-3x%29-3 Replace the second term -1x with 2x-3x.



%282x%5E2%2B2x%29%2B%28-3x-3%29 Group the terms into two pairs.



2x%28x%2B1%29%2B%28-3x-3%29 Factor out the GCF 2x from the first group.



2x%28x%2B1%29-3%28x%2B1%29 Factor out 3 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%282x-3%29%28x%2B1%29 Combine like terms. Or factor out the common term x%2B1



===============================================================



Answer:



So 2%2Ax%5E2-x-3 factors to %282x-3%29%28x%2B1%29.



In other words, 2%2Ax%5E2-x-3=%282x-3%29%28x%2B1%29.



Note: you can check the answer by expanding %282x-3%29%28x%2B1%29 to get 2%2Ax%5E2-x-3 or by graphing the original expression and the answer (the two graphs should be identical).