Problem 71 - Algebra 1: Concepts and Skills

Question

Add or subtract the polynomials. (Lesson 10.1) $$\left(16 p^{3}-p^{2}+24\right)+\left(12 p^{2}-8 p-16\right)$$

Step-by-Step Solution

Answer

The polynomial resulting from the addition is \(16 p^{3} + 8\).

Step 1: Write down the polynomials

The polynomials are \(16 p^{3}-p^{2}+24\) and \(12 p^{2}-8 p-16\). These are the expressions we will be adding.

Step 2: Combine like terms

The polynomial is \((16 p^{3} - p^{2} + 24) + (12 p^{2} - 8 p - 16)\). Begin by combining the like terms, which are the terms with the same variables and the same exponential powers. In this case, \(p^{3}\), \(p^{2}\), \(p\), and the constants (numbers) can be combined separately. Doing this calculation leads to \(16 p^{3} + 0 p^{2} + 0 p + 8\).

Step 3: Simplify the polynomial

The resulting polynomial now becomes \(16 p^{3} + 0 p^{2} + 0 p + 8\). However, it can be simplified further by omitting the terms with coefficient 0, so the final polynomial is \(16 p^{3} + 8\).

Key Concepts

Polynomial Operations

Understanding polynomial operations is crucial for mastering algebra, especially when it comes to adding polynomials. Like the task of adding \(16 p^{3}-p^{2}+24\) and \(12 p^{2}-8 p-16\), the process involves combining polynomials to form a new polynomial. To do this effectively, the first step is to identify and arrange terms with the same degree, which are often referred to as 'like terms'. Degrees in polynomials correspond to the exponent on the variable; for instance, in \(p^{3}\), the degree is 3.

When adding polynomials, simply add the coefficients (numbers in front of the variable) of like terms while keeping the variables and exponents unchanged. For any polynomial operation, it is important to keep in mind the order of operations and to handle any parentheses by first distributing any coefficients outside the parentheses if necessary.

Combine Like Terms

Combining like terms, which you'll often encounter when working with polynomials, is a method for streamlining expressions to make them simpler to work with. Like terms are terms that have exactly the same variable raised to the same power. For instance, in our exercise, the like terms are \(16 p^{3}\) from the first polynomial and there are no corresponding cubic terms in the second polynomial.

In the provided polynomials, \(16 p^{3}-p^{2}+24\) and \(12 p^{2}-8 p-16\), the like terms \(p^{2}\) terms can be combined: \( -p^{2} + 12 p^{2} = 11 p^{2}\). It's essential to keep in mind that only the coefficients of like terms are added or subtracted, not the variables or their exponents. The operation is simple addition or subtraction.

Simplify Polynomial Expressions

To simplify polynomial expressions, we take the results from combining like terms and remove any unnecessary components. Simplification can include eliminating terms with coefficients of zero, as these do not affect the value of the expression. In the given example, once you've combined the like terms, you are left with \(16 p^{3} + 11 p^{2} - 8 p + 8\). Simplifying involves identifying and omitting the zero coefficient terms, leading us to the final expression of \(16 p^{3} + 11 p^{2} - 8 p + 8\), as there are no zero coefficient terms here.

Always look for opportunities to make the expression as compact as possible without changing its meaning or value. This not only makes the mathematics look cleaner, but it can also make it easier to use in subsequent calculations or when graphing the polynomial.