Question

Add or subtract. $$\left(a^{4}-12 a\right)+\left(4 a^{3}+11 a-1\right)$$

Short answer

The simplified expression is \(a^{4} + 4a^{3} - a -1\).

Step-by-step solution

Identify Like Terms

In the given expressions \(a^{4}-12a\) and \(4a^{3}+11a-1\), identify terms involving the same powers of a. We have \(a^{4}\) in the first expression, and \(4a^{3}\) and \(-12a+11a\) as like terms.

Combine Like Terms

Add the like terms separately. This gives us \(a^{4} + 4a^{3}\) from the highest power terms of a, and \(-12a+11a\) from the terms with a to the power of 1.

Simplify the Expression

Simplify the expression further to get the final result. The term \(-12a+11a\) simplifies to \(-a\). This leads us to the final expression: \(a^{4} + 4a^{3} - a -1\).

Key concepts

Combining Like Terms

To successfully navigate polynomial addition problems like \(\left(a^{4}-12 a\right)+\left(4 a^{3}+11 a-1\right)\), the first step typically involves combining like terms. Like terms are terms that contain the same variables raised to the same powers.

Imagine you have a basket of fruits: apples pair with apples, and oranges with oranges; similarly, in algebra, we pair terms that are alike. For instance, terms with \(a^4\) will only combine with other \(a^4\) terms. Our given expressions don't have matching \(a^4\) terms, so we keep the lone \(a^4\). However, for the power of \(a^1\), we see \(\-12a\) and \(11a\). These are 'alike' and can be combined by adding their coefficients to get \(\-a\).

Why is this crucial? Combining like terms allows us to simplify the expression to its most basic form, making it easier to understand and solve. Moreover, it prevents us from getting mislead by the complexity of multiple terms and helps us focus on the essential parts of the equation.

Simplifying Expressions

Building on the foundation of combining like terms, the next logical stride is simplifying expressions. Once like terms are combined, we are left with a more manageable expression that can be further simplified. Simplification involves reducing the polynomial to as few terms as possible.

In our polynomial addition example, after combining like terms, we do not have any further simplification since there are no more like terms to combine. Therefore, our simplified expression is simply \(a^4 + 4a^3 - a - 1\).

Simplification is not merely about making the expression shorter but also about making it clearer and more accessible. It facilitates a more efficient calculation of values for the variables and can even reveal potential patterns or relationships in the algebraic structure.

Algebraic Operations

The last overarching concept in our polynomial addition journey is the execution of algebraic operations. These operations encompass addition, subtraction, multiplication, and division of algebraic expressions, themselves grounded in the same principles we use for arithmetic operations on numbers.

For our exercise, the key operation is addition. We execute it by lining up the terms that we identified as 'like' earlier and adding them together. This operation has been appropriately planned through the steps of identifying and combining like terms. Thus, we assured that only appropriate terms are brought together. Algebraic operations, because of their orderly nature, often lead to the simplification of complex-looking expressions into a more approachable and solvable state.