Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, find the sum or difference. \(\left(6 w^{2}+24 w+24\right)-\left(3 w^{2}-6 w+3\right)\)

Step-by-Step Solution

Answer

The result is \(3w^2 + 30w + 21\).

Step 1: Distribute Negative Sign

To subtract the polynomials, first distribute the negative sign across the second polynomial. This means you will change the sign of each term inside the second parentheses:\[-(3w^2 - 6w + 3) = -3w^2 + 6w - 3\]

Step 2: Rewrite the Expression

Rewrite the expression by combining the first polynomial with the result from Step 1:\[(6w^2 + 24w + 24) + (-3w^2 + 6w - 3)\]

Step 3: Combine Like Terms

Combine the like terms from the rewritten expression:- Combine \(6w^2\) and \(-3w^2\): \[6w^2 - 3w^2 = 3w^2\]- Combine \(24w\) and \(6w\): \[24w + 6w = 30w\]- Combine \(24\) and \(-3\): \[24 - 3 = 21\]Thus, the simplified expression is:\[3w^2 + 30w + 21\]

Step 4: Write the Final Expression

The final expression representing the sum or difference is:\[3w^2 + 30w + 21\]

Key Concepts

Combining Like Terms

When working with polynomials, a critical skill is the ability to combine like terms. Like terms are terms within the same expression that have the same variables raised to the same power. In the expression \(6w^2 + 24w + 24\), we have different types of terms based on the variable \(w\) and their powers.

• Terms with \(w^2\) are like terms: In this example are \(6w^2\) and \(-3w^2\).
• Terms with \(w\) are like terms: In this example are \(24w\) and \(6w\).
• Constant terms, which do not contain any variables: In this example are \(24\) and \(-3\).

By identifying like terms, you can simplify expressions by adding or subtracting their coefficients. For instance, combining \(6w^2\) and \(-3w^2\) results in \(3w^2\). This way, combining like terms helps simplify a polynomial to its most reduced form, making calculations and problem-solving much more straightforward.

Distributive Property

The distributive property is a fundamental concept in mathematics that helps in simplifying expressions, particularly when dealing with subtraction or multiplication. The property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
However, when subtracting polynomials, we need to distribute a negative sign across the terms of the polynomial being subtracted, effectively changing the sign of each term. In the exercise, the expression \( -(3w^2 - 6w + 3) \) becomes:

• \(-3w^2\)
• \(+6w\)
• \(-3\)

This sign change ensures the correct subtraction when combining with other polynomials. The distributive property is not only useful in arithmetic but also lays the groundwork for more advanced operations involving algebraic expressions.

Simplifying Expressions

Simplification in mathematics involves rewriting an expression in the most concise and clear form possible. This process can include combining like terms, applying the distributive property, and reducing complexity. By simplifying, you make expressions more manageable to work with in further calculations or operations.
In the given exercise, after distributing the negative sign to the terms of the second polynomial and rewriting the expression, we ended up with \(6w^2 + 24w + 24 + (-3w^2 + 6w - 3)\). By combining like terms (as explained in an earlier section), we achieved the simplified expression \(3w^2 + 30w + 21\).
Simplifying expressions is not just about making them shorter, but also about enhancing their clarity and making mathematical reasoning more apparent. This is especially important in algebra, where complex expressions can be greatly simplified, revealing more about the problem being solved.