Question 187131


{{{8x^3-14x^2-15x}}} Start with the given expression



{{{x(8x^2-14x-15)}}} Factor out the GCF {{{x}}}



Now let's focus on the inner expression {{{8x^2-14x-15}}}





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Looking at {{{8x^2-14x-15}}} we can see that the first term is {{{8x^2}}} and the last term is {{{-15}}} where the coefficients are 8 and -15 respectively.


Now multiply the first coefficient 8 and the last coefficient -15 to get -120. Now what two numbers multiply to -120 and add to the  middle coefficient -14? Let's list all of the factors of -120:




Factors of -120:

1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120


-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -120

(1)*(-120)

(2)*(-60)

(3)*(-40)

(4)*(-30)

(5)*(-24)

(6)*(-20)

(8)*(-15)

(10)*(-12)

(-1)*(120)

(-2)*(60)

(-3)*(40)

(-4)*(30)

(-5)*(24)

(-6)*(20)

(-8)*(15)

(-10)*(12)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-120</td><td>1+(-120)=-119</td></tr><tr><td align="center">2</td><td align="center">-60</td><td>2+(-60)=-58</td></tr><tr><td align="center">3</td><td align="center">-40</td><td>3+(-40)=-37</td></tr><tr><td align="center">4</td><td align="center">-30</td><td>4+(-30)=-26</td></tr><tr><td align="center">5</td><td align="center">-24</td><td>5+(-24)=-19</td></tr><tr><td align="center">6</td><td align="center">-20</td><td>6+(-20)=-14</td></tr><tr><td align="center">8</td><td align="center">-15</td><td>8+(-15)=-7</td></tr><tr><td align="center">10</td><td align="center">-12</td><td>10+(-12)=-2</td></tr><tr><td align="center">-1</td><td align="center">120</td><td>-1+120=119</td></tr><tr><td align="center">-2</td><td align="center">60</td><td>-2+60=58</td></tr><tr><td align="center">-3</td><td align="center">40</td><td>-3+40=37</td></tr><tr><td align="center">-4</td><td align="center">30</td><td>-4+30=26</td></tr><tr><td align="center">-5</td><td align="center">24</td><td>-5+24=19</td></tr><tr><td align="center">-6</td><td align="center">20</td><td>-6+20=14</td></tr><tr><td align="center">-8</td><td align="center">15</td><td>-8+15=7</td></tr><tr><td align="center">-10</td><td align="center">12</td><td>-10+12=2</td></tr></table>



From this list we can see that 6 and -20 add up to -14 and multiply to -120



Now looking at the expression {{{8x^2-14x-15}}}, replace {{{-14x}}} with {{{6x+-20x}}} (notice {{{6x+-20x}}} adds up to {{{-14x}}}. So it is equivalent to {{{-14x}}})


{{{8x^2+highlight(6x+-20x)+-15}}}



Now let's factor {{{8x^2+6x-20x-15}}} by grouping:



{{{(8x^2+6x)+(-20x-15)}}} Group like terms



{{{2x(4x+3)-5(4x+3)}}} Factor out the GCF of {{{2x}}} out of the first group. Factor out the GCF of {{{-5}}} out of the second group



{{{(2x-5)(4x+3)}}} Since we have a common term of {{{4x+3}}}, we can combine like terms


So {{{8x^2+6x-20x-15}}} factors to {{{(2x-5)(4x+3)}}}



So this also means that {{{8x^2-14x-15}}} factors to {{{(2x-5)(4x+3)}}} (since {{{8x^2-14x-15}}} is equivalent to {{{8x^2+6x-20x-15}}})




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So our expression goes from {{{x(8x^2-14x-15)}}} and factors further to {{{x(2x-5)(4x+3)}}}



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Answer:


So {{{8x^3-14x^2-15x}}} factors to {{{x(2x-5)(4x+3)}}}