```Question 173367
# 1

{{{3x^2y(y^2-1)}}} Factor out the GCF {{{3x^2y}}}

{{{3x^2y(y+1)(y-1)}}} Factor {{{y^2-1}}} to get {{{(y+1)(y-1)}}} (by using the difference of squares)

So {{{3x^2y^3-3x^2y}}} factors to {{{3x^2y(y+1)(y-1)}}}

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# 2

Note: the expression {{{x^2+144}}} really looks like {{{x^2+0x+144}}}

Looking at {{{x^2+0x+144}}} we can see that the first term is {{{x^2}}} and the last term is {{{144}}} where the coefficients are 1 and 144 respectively.

Now multiply the first coefficient 1 and the last coefficient 144 to get 144. Now what two numbers multiply to 144 and add to the  middle coefficient 0? Let's list all of the factors of 144:

Factors of 144:

1,2,3,4,6,8,9,12,16,18,24,36,48,72

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 144

1*144

2*72

3*48

4*36

6*24

8*18

9*16

12*12

(-1)*(-144)

(-2)*(-72)

(-3)*(-48)

(-4)*(-36)

(-6)*(-24)

(-8)*(-18)

(-9)*(-16)

(-12)*(-12)

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

Now which of these pairs add to 0? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 0

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">144</td><td>1+144=145</td></tr><tr><td align="center">2</td><td align="center">72</td><td>2+72=74</td></tr><tr><td align="center">3</td><td align="center">48</td><td>3+48=51</td></tr><tr><td align="center">4</td><td align="center">36</td><td>4+36=40</td></tr><tr><td align="center">6</td><td align="center">24</td><td>6+24=30</td></tr><tr><td align="center">8</td><td align="center">18</td><td>8+18=26</td></tr><tr><td align="center">9</td><td align="center">16</td><td>9+16=25</td></tr><tr><td align="center">12</td><td align="center">12</td><td>12+12=24</td></tr><tr><td align="center">-1</td><td align="center">-144</td><td>-1+(-144)=-145</td></tr><tr><td align="center">-2</td><td align="center">-72</td><td>-2+(-72)=-74</td></tr><tr><td align="center">-3</td><td align="center">-48</td><td>-3+(-48)=-51</td></tr><tr><td align="center">-4</td><td align="center">-36</td><td>-4+(-36)=-40</td></tr><tr><td align="center">-6</td><td align="center">-24</td><td>-6+(-24)=-30</td></tr><tr><td align="center">-8</td><td align="center">-18</td><td>-8+(-18)=-26</td></tr><tr><td align="center">-9</td><td align="center">-16</td><td>-9+(-16)=-25</td></tr><tr><td align="center">-12</td><td align="center">-12</td><td>-12+(-12)=-24</td></tr></table>

None of these pairs of factors add to 0. So the expression {{{x^2+0x+144}}} cannot be factored

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# 3

Looking at the expression {{{5x^2-14x+16}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-14}}}, and the last term is {{{16}}}.

Now multiply the first coefficient {{{5}}} by the last term {{{16}}} to get {{{(5)(16)=80}}}.

Now the question is: what two whole numbers multiply to {{{80}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-14}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{80}}} (the previous product).

Factors of {{{80}}}:

1,2,4,5,8,10,16,20,40,80

-1,-2,-4,-5,-8,-10,-16,-20,-40,-80

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

These factors pair up and multiply to {{{80}}}.

1*80
2*40
4*20
5*16
8*10
(-1)*(-80)
(-2)*(-40)
(-4)*(-20)
(-5)*(-16)
(-8)*(-10)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-14}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>80</font></td><td  align="center"><font color=black>1+80=81</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>40</font></td><td  align="center"><font color=black>2+40=42</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>20</font></td><td  align="center"><font color=black>4+20=24</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>16</font></td><td  align="center"><font color=black>5+16=21</font></td></tr><tr><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>10</font></td><td  align="center"><font color=black>8+10=18</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-80</font></td><td  align="center"><font color=black>-1+(-80)=-81</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-40</font></td><td  align="center"><font color=black>-2+(-40)=-42</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-20</font></td><td  align="center"><font color=black>-4+(-20)=-24</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>-16</font></td><td  align="center"><font color=black>-5+(-16)=-21</font></td></tr><tr><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>-8+(-10)=-18</font></td></tr></table>

From the table, we can see that there are no pairs of numbers which add to {{{-14}}}.

So {{{5x^2-14x+16}}} cannot be factored.

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# 4

Looking at the expression {{{2x^2+11x-9}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{11}}}, and the last term is {{{-9}}}.

Now multiply the first coefficient {{{2}}} by the last term {{{-9}}} to get {{{(2)(-9)=-18}}}.

Now the question is: what two whole numbers multiply to {{{-18}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{11}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-18}}} (the previous product).

Factors of {{{-18}}}:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18

Note: list the negative of each factor. 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)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{11}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>1+(-18)=-17</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>2+(-9)=-7</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>3+(-6)=-3</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-1+18=17</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>-2+9=7</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-3+6=3</font></td></tr></table>

From the table, we can see that there are no pairs of numbers which add to {{{11}}}. So {{{2x^2+11x-9}}} cannot be factored.

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Note: there's always the possibility that you cannot factor an expression. However, most books will only throw them out in small doses. So I would double check your problems to make sure that you copied them down correctly.```