Need help on Algebra 1 hw?

I can't figure out how to solve for p, 3x^p(4x^{2p+3}+2x^{3p-2})=12x^{12}+6x^{10}

1 Answer
R. Nguyen
Jan 30, 2018

#p = 3#

Explanation:

#color(red)(3x^p)(4x^{2p+3}+2x^{3p-2})=12x^{12}+6x^{10}#

First, we might want to expand out our left-hand side.

#color(red)(3x^p)*4x^{2p+3}+color(red)(3x^p)*2x^{3p-2}=12x^{12}+6x^{10}#

Recall that #x^a*x^b=x^{a+b}#.

#(color(red)3 * 4)x^{color(red)p + 2p + 3} + (color(red)3 * 2)x^{color(red)p + 3p - 2} = 12x^{12}+6x^{10}#

#12x^{color(blue)(3p + 3)} + 6x^{color(blue)(4p - 2)} = 12x^{color(blue)12}+6x^{color(blue)10}#

Doing a little bit of pattern matching, we notice that #color(blue)(3p + 3)# corresponds exactly to #color(blue)12#, and that #color(blue)(4p-2)# corresponds exactly to #color(blue)(10)#. This means that they must be equal.

#3p + 3 = 12#

#4p - 2 = 10#

Solving either of these equations tells us that #p = 3#.

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