find p such that 3x^p(4x^2p+^3+2x^3p-2)=12x^12+6x^10
To find the value of p such that the equation 3x^p(4x^2p+3 + 2x^3p-2) = 12x^12 + 6x^10 holds, we need to simplify both sides of the equation and solve for p. Here's how you can do it step by step:
1. First, simplify the expressions on both sides of the equation:
3x^p(4x^2p+3 + 2x^3p-2) = 12x^12 + 6x^10
3x^p(4x^(2p+3) + 2/x^(2-p)) = 12x^12 + 6x^10
2. Next, expand the expressions on both sides of the equation:
12x^(p+2p+3) + 6/x^(p-2p) = 12x^12 + 6x^10
3. Simplify the exponents on both sides of the equation:
12x^(3p+3) + 6/x^(-p) = 12x^12 + 6x^10
4. Now, let's compare the coefficients and exponents on both sides of the equation:
Coefficients: 12x^(3p+3) and 12x^12
Exponents: 3p+3 and 12
Since the coefficients are equal, we can equate the exponents:
3p + 3 = 12
5. Solve for p:
Subtract 3 from both sides of the equation:
3p = 12 - 3
3p = 9
Divide both sides of the equation by 3:
p = 9/3
6. Simplify p:
p = 3
Therefore, the value of p that satisfies the equation is p = 3.