Problem 108 - Intermediate Algebra

Question

Perform each operation. $$ \left(4 u^{2}+z^{2}-3 u^{2} z^{2}\right)-\left(u^{3}+3 z^{2}-3 u^{2} z^{2}\right) $$

Step-by-Step Solution

Answer

The result is \\(-u^{3} + 4u^{2} - 2z^{2}\\)."

Step 1: Distribute the Negative Sign

First, remove the parentheses by distributing the negative sign in front of \(\left(u^{3}+3z^{2}-3u^{2}z^{2}\right)\). This changes every term inside the parentheses to its opposite. So, we have: \(4u^{2} + z^{2} - 3u^{2}z^{2} - u^{3} - 3z^{2} + 3u^{2}z^{2}\).

Step 2: Simplify by Combining Like Terms

Combine like terms from the expression obtained after distribution. The like terms are grouped as follows:- \(4u^{2}\) and \(- u^{3}\) are not like terms (different powers of \('u'\)).- \(z^{2}\) and \(-3z^{2}\).- \(-3u^{2}z^{2}\) and \(+3u^{2}z^{2}\).Combine these like terms: - \(4u^{2}\) remains unchanged.- \(-u^{3}\) remains unchanged.- For \(z^{2}\) terms: \(z^{2} - 3z^{2} = -2z^{2}\).- \(-3u^{2}z^{2}\) and \(+3u^{2}z^{2} = 0\) (they cancel each other out).The simplified expression is: \(-u^{3} + 4u^{2} - 2z^{2}\).

Key Concepts

Distributing Negative Sign

When subtracting polynomials, one of the crucial steps is distributing the negative sign across the polynomial being subtracted. This means that every term inside the parentheses is affected. Let’s break this down:

• If a term is positive, it becomes negative when the negative sign is applied.
• If a term is negative, it becomes positive.

So when you have \(\left(a + b + c\right) - \left(x - y + z\right)\), you distribute the negative sign to get: \(a + b + c - x + y - z\). Each term is reversed in its sign, which is critical to getting the right subtraction result. Understanding this helps avoid common mistakes in polynomial subtraction.

Combining Like Terms

After distributing any negative signs, the next step in simplifying polynomials is combining like terms. Like terms are terms that contain the same variables raised to the same power. For example:

• In the expression \(3x^2 + 5x^2 - x + 4\), the terms \(3x^2\) and \(5x^2\) are like terms because they both contain \(x^2\).
• When combining like terms, add or subtract the coefficients but keep the variable part unchanged.

Let’s apply this to \(4u^{2} + z^{2} - 3u^{2}z^{2} - u^{3} - 3z^{2} + 3u^{2}z^{2}\).Since \(-3u^{2}z^{2}\) and \(+3u^{2}z^{2}\) cancel each other out with zero net result, you focus on the remaining like pairs. This streamlines the expression significantly. Overall, combining like terms reduces complexity and clarifies the polynomial structure.

Simplifying Polynomials

The final phase in handling polynomial expressions is simplifying them. This involves:

• Distributing negative signs correctly to open up expressions.
• Combining like terms to eliminate redundancies.
• Rewriting the expression in a more compact and clean form.

In this specific problem, we simplify \(-u^{3} + 4u^{2} - 2z^{2}\). After ensuring no more like terms remain, review your polynomial to see if there are any factoring opportunities or further cancellations.Polynomial simplification is about making the expression as neat as possible so it accurately reflects the simplest mathematical reality. This clarity is not just about solving current problems—it also aids greatly in future algebraic manipulations and calculations. A polished polynomial will make your mathematical journey much smoother.