Question 156977


{{{16z^5+12z^4-10z^3}}} Start with the given expression



{{{2z^3(8z^2+6z-5)}}} Factor out the GCF {{{2z^3}}}



Now let's focus on the inner expression {{{8z^2+6z-5}}}





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Looking at {{{8z^2+6z-5}}} we can see that the first term is {{{8z^2}}} and the last term is {{{-5}}} where the coefficients are 8 and -5 respectively.


Now multiply the first coefficient 8 and the last coefficient -5 to get -40. Now what two numbers multiply to -40 and add to the  middle coefficient 6? Let's list all of the factors of -40:




Factors of -40:

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


-1,-2,-4,-5,-8,-10,-20,-40 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -40

(1)*(-40)

(2)*(-20)

(4)*(-10)

(5)*(-8)

(-1)*(40)

(-2)*(20)

(-4)*(10)

(-5)*(8)


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



Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6



<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>-40</font></td><td  align="center"><font color=black>1+(-40)=-39</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-20</font></td><td  align="center"><font color=black>2+(-20)=-18</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>4+(-10)=-6</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>5+(-8)=-3</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>40</font></td><td  align="center"><font color=black>-1+40=39</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>20</font></td><td  align="center"><font color=black>-2+20=18</font></td></tr><tr><td  align="center"><font color=red>-4</font></td><td  align="center"><font color=red>10</font></td><td  align="center"><font color=red>-4+10=6</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>-5+8=3</font></td></tr></table>







From this list we can see that -4 and 10 add up to 6 and multiply to -40



Now looking at the expression {{{8z^2+6z-5}}}, replace {{{6z}}} with {{{-4z+10z}}} (notice {{{-4z+10z}}} adds up to {{{6z}}}. So it is equivalent to {{{6z}}})


{{{8z^2+highlight(-4z+10z)+-5}}}



Now let's factor {{{8z^2-4z+10z-5}}} by grouping:



{{{(8z^2-4z)+(10z-5)}}} Group like terms



{{{4z(2z-1)+5(2z-1)}}} Factor out the GCF of {{{4z}}} out of the first group. Factor out the GCF of {{{5}}} out of the second group



{{{(4z+5)(2z-1)}}} Since we have a common term of {{{2z-1}}}, we can combine like terms


So {{{8z^2-4z+10z-5}}} factors to {{{(4z+5)(2z-1)}}}



So this also means that {{{8z^2+6z-5}}} factors to {{{(4z+5)(2z-1)}}} (since {{{8z^2+6z-5}}} is equivalent to {{{8z^2-4z+10z-5}}})




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So our expression goes from {{{2z^3(8z^2+6z-5)}}} and factors further to {{{2z^3(4z+5)(2z-1)}}}



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Answer:



So {{{16z^5+12z^4-10z^3}}} completely factors to {{{2z^3(4z+5)(2z-1)}}}