SOLUTION: 1)6b^2-7b-5 Factor 2)m^2+4mn+4n^2 Factor 3)2a^2-13a+15 Factor 4)z^3+9z^2+18z Is this correct? z(z^2+9z+18) 5)x^3+125 Factor 6)a^

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 1)6b^2-7b-5 Factor 2)m^2+4mn+4n^2 Factor 3)2a^2-13a+15 Factor 4)z^3+9z^2+18z Is this correct? z(z^2+9z+18) 5)x^3+125 Factor 6)a^      Log On


   

Question 148867: 1)6b^2-7b-5 Factor


2)m^2+4mn+4n^2 Factor


3)2a^2-13a+15 Factor



4)z^3+9z^2+18z

Is this correct? z(z^2+9z+18)


5)x^3+125 Factor



6)a^4-ab^3





Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1



Looking at the expression 6b%5E2-7b-5, we can see that the first coefficient is 6, the second coefficient is -7, and the last term is -5.


Now multiply the first coefficient 6 by the last term -5 to get %286%29%28-5%29=-30.


Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient -7?


To find these two numbers, we need to list all of the factors of -30 (the previous product).


Factors of -30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


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


These factors pair up and multiply to -30. For instance, 1%2A30=-30, 2%2A15=-30, etc.


Since -30 is negative, this means that one factor is positive and one is negative.


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -7:


First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1



From the table, we can see that the two numbers 3 and -10 add to -7 (the middle coefficient).


So the two numbers 3 and -10 both multiply to -30 and add to -7


Now replace the middle term -7b with 3b-10b. Remember, 3 and -10 add to -7. So this shows us that 3b-10b=-7b.


6b%5E2%2Bhighlight%283b-10b%29-5 Replace the second term -7b with 3b-10b.


%286b%5E2%2B3b%29%2B%28-10b-5%29 Group the terms into two pairs.


3b%282b%2B1%29%2B%28-10b-5%29 Factor out the GCF 3b from the first group.


3b%282b%2B1%29-5%282b%2B1%29 Factor out 5 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.


%283b-5%29%282b%2B1%29 Combine like terms. Or factor out the common term 2b%2B1

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Answer:


So 6b%5E2-7b-5 factors to %283b-5%29%282b%2B1%29.


Note: you can check the answer by FOILing %283b-5%29%282b%2B1%29 to get 6b%5E2-7b-5 or by graphing the original expression and the answer (the two graphs should be identical).







# 2



Looking at 1m%5E2%2B4mn%2B4n%5E2 we can see that the first term is 1m%5E2 and the last term is 4n%5E2 where the coefficients are 1 and 4 respectively.

Now multiply the first coefficient 1 and the last coefficient 4 to get 4. Now what two numbers multiply to 4 and add to the middle coefficient 4? Let's list all of the factors of 4:



Factors of 4:
1,2

-1,-2 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 4
1*4
2*2
(-1)*(-4)
(-2)*(-2)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4

First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4



From this list we can see that 2 and 2 add up to 4 and multiply to 4


Now looking at the expression 1m%5E2%2B4mn%2B4n%5E2, replace 4mn with 2mn%2B2mn (notice 2mn%2B2mn adds up to 4mn. So it is equivalent to 4mn)

1m%5E2%2Bhighlight%282mn%2B2mn%29%2B4n%5E2


Now let's factor 1m%5E2%2B2mn%2B2mn%2B4n%5E2 by grouping:


%281m%5E2%2B2mn%29%2B%282mn%2B4n%5E2%29 Group like terms


m%28m%2B2n%29%2B2n%28m%2B2n%29 Factor out the GCF of m out of the first group. Factor out the GCF of 2n out of the second group


%28m%2B2n%29%28m%2B2n%29 Since we have a common term of m%2B2n, we can combine like terms

So 1m%5E2%2B2mn%2B2mn%2B4n%5E2 factors to %28m%2B2n%29%28m%2B2n%29


So this also means that 1m%5E2%2B4mn%2B4n%5E2 factors to %28m%2B2n%29%28m%2B2n%29 (since 1m%5E2%2B4mn%2B4n%5E2 is equivalent to 1m%5E2%2B2mn%2B2mn%2B4n%5E2)


note: %28m%2B2n%29%28m%2B2n%29 is equivalent to %28m%2B2n%29%5E2 since the term m%2B2n occurs twice. So 1m%5E2%2B4mn%2B4n%5E2 also factors to %28m%2B2n%29%5E2



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Answer:
So m%5E2%2B4mn%2B4n%5E2 factors to %28m%2B2n%29%5E2





# 3



Looking at the expression 2a%5E2-13a%2B15, we can see that the first coefficient is 2, the second coefficient is -13, and the last term is 15.


Now multiply the first coefficient 2 by the last term 15 to get %282%29%2815%29=30.


Now the question is: what two whole numbers multiply to 30 (the previous product) and add to the second coefficient -13?


To find these two numbers, we need to list all of the factors of 30 (the previous product).


Factors of 30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


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


These factors pair up and multiply to 30. For instance, 1%2A30=30, 2%2A15=30, etc.


Since 30 is positive, this means that either
a) both factors are positive, or...
b) both factors are negative.


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -13:


First NumberSecond NumberSum
1301+30=31
2152+15=17
3103+10=13
565+6=11
-1-30-1+(-30)=-31
-2-15-2+(-15)=-17
-3-10-3+(-10)=-13
-5-6-5+(-6)=-11



From the table, we can see that the two numbers -3 and -10 add to -13 (the middle coefficient).


So the two numbers -3 and -10 both multiply to 30 and add to -13


Now replace the middle term -13a with -3a-10a. Remember, -3 and -10 add to -13. So this shows us that -3a-10a=-13a.


2a%5E2%2Bhighlight%28-3a-10a%29%2B15 Replace the second term -13a with -3a-10a.


%282a%5E2-3a%29%2B%28-10a%2B15%29 Group the terms into two pairs.


a%282a-3%29%2B%28-10a%2B15%29 Factor out the GCF a from the first group.


a%282a-3%29-5%282a-3%29 Factor out 5 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.


%28a-5%29%282a-3%29 Combine like terms. Or factor out the common term 2a-3

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


Answer:


So 2a%5E2-13a%2B15 factors to %28a-5%29%282a-3%29.


Note: you can check the answer by FOILing %28a-5%29%282a-3%29 to get 2a%5E2-13a%2B15 or by graphing the original expression and the answer (the two graphs should be identical).






# 4



z%5E3%2B9z%5E2%2B18z Start with the given expression


z%28z%5E2%2B9z%2B18%29 Factor out the GCF z


Now let's focus on the inner expression z%5E2%2B9z%2B18




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Looking at 1z%5E2%2B9z%2B18 we can see that the first term is 1z%5E2 and the last term is 18 where the coefficients are 1 and 18 respectively.

Now multiply the first coefficient 1 and the last coefficient 18 to get 18. Now what two numbers multiply to 18 and add to the middle coefficient 9? Let's list all of the factors of 18:



Factors of 18:
1,2,3,6,9,18

-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 18
1*18
2*9
3*6
(-1)*(-18)
(-2)*(-9)
(-3)*(-6)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 9

First NumberSecond NumberSum
1181+18=19
292+9=11
363+6=9
-1-18-1+(-18)=-19
-2-9-2+(-9)=-11
-3-6-3+(-6)=-9



From this list we can see that 3 and 6 add up to 9 and multiply to 18


Now looking at the expression 1z%5E2%2B9z%2B18, replace 9z with 3z%2B6z (notice 3z%2B6z adds up to 9z. So it is equivalent to 9z)

1z%5E2%2Bhighlight%283z%2B6z%29%2B18


Now let's factor 1z%5E2%2B3z%2B6z%2B18 by grouping:


%281z%5E2%2B3z%29%2B%286z%2B18%29 Group like terms


z%28z%2B3%29%2B6%28z%2B3%29 Factor out the GCF of z out of the first group. Factor out the GCF of 6 out of the second group


%28z%2B6%29%28z%2B3%29 Since we have a common term of z%2B3, we can combine like terms

So 1z%5E2%2B3z%2B6z%2B18 factors to %28z%2B6%29%28z%2B3%29


So this also means that 1z%5E2%2B9z%2B18 factors to %28z%2B6%29%28z%2B3%29 (since 1z%5E2%2B9z%2B18 is equivalent to 1z%5E2%2B3z%2B6z%2B18)



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So our expression goes from z%28z%5E2%2B9z%2B18%29 and factors further to z%28z%2B6%29%28z%2B3%29


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Answer:

So z%5E3%2B9z%5E2%2B18z factors to z%28z%2B6%29%28z%2B3%29





# 5



x%5E3%2B125 Start with the given expression.


%28x%29%5E3%2B%285%29%5E3 Rewrite x%5E3 as %28x%29%5E3. Rewrite 125 as %285%29%5E3.


%28x%2B5%29%28%28x%29%5E2-%28x%29%285%29%2B%285%29%5E2%29 Now factor by using the sum of cubes formula. Remember the sum of cubes formula is A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29


%28x%2B5%29%28x%5E2-5x%2B25%29 Multiply

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Answer:

So x%5E3%2B125 factors to %28x%2B5%29%28x%5E2-5x%2B25%29.

In other words, x%5E3%2B125=%28x%2B5%29%28x%5E2-5x%2B25%29





# 6


a%5E4-ab%5E3 Start with the given expression


a%28a%5E3-b%5E3%29 Factor out the GCF a


Now let's focus on the inner expression a%5E3-b%5E3

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%28a%29%5E3-%28b%29%5E3 Rewrite a%5E3 as %28a%29%5E3. Rewrite b%5E3 as %28b%29%5E3.


%28a-b%29%28%28a%29%5E2%2B%28a%29%28b%29%2B%28b%29%5E2%29 Now factor by using the difference of cubes formula. Remember the difference of cubes formula is A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29


%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29 Multiply



So a%5E3-b%5E3 factors to %28a-b%29%28a%5E2%2Bab%2Bb%5E2%29.



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Answer:

So a%5E4-b%5E3 factors to a%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29.