Question

Add or subtract. $$\left(16 p^{3}-p^{2}+24\right)+\left(12 p^{2}-8 p-16\right)$$

Short answer

The result is: \(16p^3 + 13p^2 - 8p + 8\).

Step-by-step solution

Identify Like Terms

Looking at \(16 p^{3}-p^{2}+24\)+ \(12 p^{2}-8 p-16\), the like terms are terms that have the same variable and the same exponent. Thus they are: \(16p^3\) and \(p^2\) from the first trinomial, and \(12p^2\) and \(-8p\) from the second trinomial.

Add or subtract the Like Terms

Now, let's add or subtract each pair of like terms: \n\nFor \(16p^3\), there is no corresponding term in the second polynomial, so it remains as is. \n\n\(p^2\) and \(12p^2\) are like terms as they have the same variable (p) and exponent (2). By adding these, we get \(p^2 + 12p^2 = 13p^2\). \n\nThe terms 24 and -16 in both polynomials are the constants. By adding these, we get \(24 - 16 = 8\). Then, we write \( -8p \) as it is in the second polynomial.\n\nSo, the result of the addition will be \(16p^3 + 13p^2 - 8p + 8\)

Write the result in standard form

The standard form of a polynomial orders the terms from highest degree to lowest. In this case the result will be \(16p^3 + 13p^2 - 8p + 8\). This is the final answer of this exercise.