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Chapter 5: Problem 13

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

Short Answer

Expert verified
7u^3 - 8u + 9

Step by step solution

01

Write the expression

Start with the given expression to be divided: \ \ \( \frac{56u^5 - 64u^3 + 72u^2}{8u^2} \)
02

Break it into separate terms

Divide each term in the numerator by the denominator separately: \ \ \( \frac{56u^5}{8u^2} - \frac{64u^3}{8u^2} + \frac{72u^2}{8u^2} \)
03

Simplify each term

Calculate the division for each term: \ \ \( \frac{56u^5}{8u^2} = 7u^3 \) \( \frac{64u^3}{8u^2} = 8u \) \( \frac{72u^2}{8u^2} = 9 \)
04

Put it all together

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Algebra
Algebra can sometimes seem daunting, but it鈥檚 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.

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Most popular questions from this chapter

\(\left(h^{8}-4 h^{2}+100 h+20\right) \div\left(4 h^{2}\right)\)

A square is a rectangle in which the lengths of all four sides are equal. If \(s=\) length of a side, write a polynomial expression in \(s\) that represents the area.

\(\left(12 y^{3}-10 y^{2}+6 y\right) \div(5 y)\)

The pattern for the difference of squares is given as \((a-b)(a+b)=a^{2}-b^{2}\). Is this equivalent to the pattern \((a+b)(a-b)=a^{2}-b^{2} ?\) Explain.

The width of a rectangle is \(10 \mathrm{~cm}\) shorter than the length. a. If \(L=\) length, write a polynomial expression in \(L\) that represents the width, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(L\) that represents the perimeter. c. Write a polynomial expression in \(L\) that represents the area.
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