Question 281637

Looking at {{{5p^2+14pq-3q^2}}} we can see that the first term is {{{5p^2}}} and the last term is {{{-3q^2}}} where the coefficients are 5 and -3 respectively.


Now multiply the first coefficient 5 and the last coefficient -3 to get -15. Now what two numbers multiply to -15 and add to the  middle coefficient 14? Let's list all of the factors of -15:




Factors of -15:

1,3,5,15


-1,-3,-5,-15 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -15

(1)*(-15)

(3)*(-5)

(-1)*(15)

(-3)*(5)


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"><font color=black>1</font></td><td  align="center"><font color=black>-15</font></td><td  align="center"><font color=black>1+(-15)=-14</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>3+(-5)=-2</font></td></tr><tr><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>15</font></td><td  align="center"><font color=red>-1+15=14</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-3+5=2</font></td></tr></table>





From this list we can see that -1 and 15 add up to 14 and multiply to -15



Now looking at the expression {{{5p^2+14pq-3q^2}}}, replace {{{14pq}}} with {{{-pq+15pq}}} (notice {{{-pq+15pq}}} adds up to {{{14pq}}}. So it is equivalent to {{{14pq}}})


{{{5p^2+highlight(-pq+15pq)-3q^2}}}



Now let's factor {{{5p^2-pq+15pq-3q^2}}} by grouping:



{{{(5p^2-pq)+(15pq-3q^2)}}} Group like terms



{{{p(5p-q)+3q(5p-q)}}} Factor out the GCF of {{{p}}} out of the first group. Factor out the GCF of {{{3q}}} out of the second group



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


So {{{5p^2-pq+15pq-3q^2}}} factors to {{{(p+3q)(5p-q)}}}



So this also means that {{{5p^2+14pq-3q^2}}} factors to {{{(p+3q)(5p-q)}}} (since {{{5p^2+14pq-3q^2}}} is equivalent to {{{5p^2-pq+15pq-3q^2}}})




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     Answer:

So {{{5p^2+14pq-3q^2}}} factors to {{{(p+3q)(5p-q)}}}



In other words, {{{5p^2+14pq-3q^2=(p+3q)(5p-q)}}}