Problem 36 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ \left(-4 a^{2}-a b+15 b^{2}\right)+\left(5 a^{2}-b^{2}\right) $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(a^2 - ab + 14b^2\).

Step 1: Identify like terms

Look at the given polynomials, \[(-4a^2 - ab + 15b^2) + (5a^2 - b^2),\] and identify the like terms. Like terms have the same variables raised to the same power.

Step 2: Group the like terms

Group the like terms from the polynomials to make adding easier. The terms to group are as follows:- Terms with \(a^2\): \(-4a^2\) and \(5a^2\).- Terms with \(ab\): \(-ab\).- Terms with \(b^2\): \(15b^2\) and \(-b^2\).

Step 3: Add the coefficients of like terms

Add together the coefficients of each group of like terms:- For \(a^2\) terms: \(-4 + 5 = 1\), resulting in \(1a^2\).- There is only one \(ab\) term: \(-ab\), which remains as it is.- For \(b^2\) terms: \(15 - 1 = 14\), resulting in \(14b^2\).

Step 4: Write the simplified polynomial

Combine the results from the addition of like terms to write the final simplified polynomial expression:\[a^2 - ab + 14b^2.\]

Key Concepts

Like Terms

In polynomial mathematics, one of the foundational concepts is identifying and working with "like terms." Like terms are terms within a polynomial that contain the same variables elevated to the same powers. For example, in the polynomial \[(-4a^2 - ab + 15b^2) + (5a^2 - b^2),\]like terms have identical variable parts:

• \(a^2\) terms like \(-4a^2\) and \(5a^2\)
• \(b^2\) terms like \(15b^2\) and \(-b^2\)

By focusing on the variable components, you can identify which terms are like one another. This process is a crucial step because it sets the stage for effective polynomial addition. Recognizing they are similar because of the variable structure helps simplify the expression effectively. When you find a term like \(-ab\) which stands alone with no counterpart, it stays unchanged in the simplification process.

Coefficient Addition

Once like terms have been identified in a polynomial, the next step is about adding together their coefficients. Each term in a polynomial is composed of a coefficient (a number) and a variable part. Think of the coefficient as the "weight" of its corresponding variable term. For instance, when we examine the terms \(-4a^2\) and \(5a^2\), their coefficients are \(-4\) and \(5\) respectively. Adding these coefficients yields the simplification of\[-4 + 5 = 1\].This results in the term \(1a^2\) or simply \(a^2\).

• Notice how the variable part remains consistent across like terms and only coefficients are subject to arithmetic operations.

Similarly, for the terms \(15b^2\) and \(-b^2\), the coefficient addition is:\[15 - 1 = 14\],leading to \(14b^2\). When a term like \(-ab\) appears alone without another like term, its coefficient remains unaltered in the final simplified expression.

Simplifying Polynomials

The ultimate goal of recognizing like terms and performing coefficient addition is to simplify polynomials to their most concise form. Simplifying polynomials involves consolidating all calculations into a streamlined expression that is easier to work with. As you put this into practice with the example,\[(-4a^2 - ab + 15b^2) + (5a^2 - b^2),\]you end up combining the results:\[a^2 - ab + 14b^2.\]This expression is much cleaner than its initial form. Here are key points to remember while simplifying:

• Ensure all like terms are accurately grouped and coefficients are correctly added.
• A term without a like term such as \(-ab\) is simply carried over to the final expression as it is.

By executing these steps, you achieve a simplified polynomial where each term appears distinct and unencumbered by excess calculations. This facilitates easier manipulation or further operations in subsequent mathematical queries.