Problem 13 - Elementary Algebra

Question

\(\left(56 u^{5}-64 u^{3}+72 u^{2}\right) \div\left(8 u^{2}\right)\)

Step-by-Step Solution

Answer

7u^3 - 8u + 9

Step 1: Write the expression

Start with the given expression to be divided: \ \ \( \frac{56u^5 - 64u^3 + 72u^2}{8u^2} \)

Step 2: Break it into separate terms

Divide each term in the numerator by the denominator separately: \ \ \( \frac{56u^5}{8u^2} - \frac{64u^3}{8u^2} + \frac{72u^2}{8u^2} \)

Step 3: Simplify each term

Calculate the division for each term: \ \ \( \frac{56u^5}{8u^2} = 7u^3 \) \( \frac{64u^3}{8u^2} = 8u \) \( \frac{72u^2}{8u^2} = 9 \)

Step 4: Put it all together

Combine the simplified terms back into one expression: \ \ \( 7u^3 - 8u + 9 \)

Key Concepts

Understanding Algebra

Algebra can sometimes seem daunting, but it’s a set of tools that help us solve problems involving unknown variables. In algebra, we use letters (like u, x, y) to represent numbers we don't know yet. When dividing polynomials, as in our exercise, these letters are manipulated according to specific rules. This can simplify complex expressions and make them easier to work with. The goal is to find an equivalent form of the polynomial that's easier to understand and solve. Breaking down the problem into smaller, manageable steps, as shown in the exercise, is key to mastering algebra.

Working with Polynomials

Polynomials are expressions made up of variables raised to various powers, multiplied by coefficients, and added together. For example, in the exercise provided, we start with the polynomial \(56u^5 - 64u^3 + 72u^2\). Each term consists of a coefficient (like 56) and a power of the variable (like \(u^5\)). To simplify polynomials, we need to understand how to perform operations like addition, subtraction, multiplication, and division on these terms. The main idea is treating each term individually before combining the results at the end. This way, even complex polynomials can be broken down into simpler parts and managed more easily.

Simplifying Expressions

Simplifying polynomial expressions means making them easier to work with without changing their value. When we divided \(56u^5 - 64u^3 + 72u^2\) by \(8u^2\), we simplified it to \(7u^3 - 8u + 9\). Here's a step-by-step guide to understanding this process:

• First, divide each term in the numerator by the term in the denominator separately. For example, \(\frac{56u^5}{8u^2}\).


• Then, simplify each fraction. This involves reducing the coefficients (56 divided by 8 equals 7) and subtracting the exponents of the variables (\(u^5\) divided by \(u^2\) subtracts the exponents to give \(u^3\)).


• After simplifying each term, combine them back into a single expression. This operation simplifies the original polynomial and makes it easier to handle further calculations or to understand its behavior.

These steps can make even complex polynomials much more approachable, helping you see the patterns and solutions more clearly.