Question 148119
*[Tex \LARGE 2x^{\frac{2}{3}} - 5x^{\frac{1}{3}} - 3] Start with the given expression.



Let *[Tex \LARGE z=x^{\frac{1}{3}}]. This means that *[Tex \LARGE z^{2}=\left(x^{\frac{1}{3}}\right)^{2}=x^{\frac{2}{3}}]



*[Tex \LARGE 2z^{2} - 5z - 3] Replace *[Tex \LARGE x^{\frac{2}{3}}] with *[Tex \LARGE z^{2}]. Replace *[Tex \LARGE x^{\frac{1}{3}}] with *[Tex \LARGE z]




Looking at {{{2z^2-5z-3}}} we can see that the first term is {{{2z^2}}} and the last term is {{{-3}}} where the coefficients are 2 and -3 respectively.


Now multiply the first coefficient 2 and the last coefficient -3 to get -6. Now what two numbers multiply to -6 and add to the  middle coefficient -5? Let's list all of the factors of -6:




Factors of -6:

1,2,3,6


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


These factors pair up and multiply to -6

(1)*(-6)

(2)*(-3)

(-1)*(6)

(-2)*(3)


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



Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-6</td><td>1+(-6)=-5</td></tr><tr><td align="center">2</td><td align="center">-3</td><td>2+(-3)=-1</td></tr><tr><td align="center">-1</td><td align="center">6</td><td>-1+6=5</td></tr><tr><td align="center">-2</td><td align="center">3</td><td>-2+3=1</td></tr></table>



From this list we can see that 1 and -6 add up to -5 and multiply to -6



Now looking at the expression {{{2z^2-5z-3}}}, replace {{{-5z}}} with {{{1z+-6z}}} (notice {{{1z+-6z}}} adds up to {{{-5z}}}. So it is equivalent to {{{-5z}}})


{{{2z^2+highlight(1z+-6z)+-3}}}



Now let's factor {{{2z^2+z-6z-3}}} by grouping:



{{{(2z^2+z)+(-6z-3)}}} Group like terms



{{{z(2z+1)-3(2z+1)}}} Factor out the GCF of {{{z}}} out of the first group. Factor out the GCF of {{{-3}}} out of the second group



{{{(z-3)(2z+1)}}} Since we have a common term of {{{2z+1}}}, we can combine like terms



So {{{2z^2-5z-3}}} factors to {{{(z-3)(2z+1)}}}



*[Tex \LARGE \left(x^{\frac{1}{3}}-3\right)\left(2x^{\frac{1}{3}}+1\right)] Now replace "z" with *[Tex \LARGE x^{\frac{1}{3}}] 




So *[Tex \LARGE 2x^{\frac{2}{3}} - 5x^{\frac{1}{3}} - 3] factors to *[Tex \LARGE \left(x^{\frac{1}{3}}-3\right)\left(2x^{\frac{1}{3}}+1\right)]



In other words, *[Tex \LARGE 2x^{\frac{2}{3}} - 5x^{\frac{1}{3}} - 3=\left(x^{\frac{1}{3}}-3\right)\left(2x^{\frac{1}{3}}+1\right)] where every variable is positive.