Question

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}

Answer 1

`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`.