Problem 84 - Algebra A Combined Function

Question

Match each expression on the left with its simplification on the right. Not all letters on the right must be used and a letter may be used more than once. a. \(3 y\) b. \(9 y-6 y^{2}\) c. \(10 x\) d. \(25 x^{2}\) e. \(10 x-6\) f. none of these $$ 5 x+5 x $$

Step-by-Step Solution

Answer

Expression c: \(10x\) matches the simplified expression \(5x + 5x\).

Step 1: Understand the Expressions

Before matching, first take a look at each expression given: - Left expressions: (a) \(3y\), (b) \(9y - 6y^2\), (c) \(10x\), (d) \(25x^2\), (e) \(10x - 6\), and (f) 'none of these'.- Right expression not included but question asks to match with \(5x + 5x\).

Step 2: Simplify Right Expression

The expression to simplify is \(5x + 5x\). Combine like terms by adding the coefficients of \(x\) together: \[ 5x + 5x = (5+5)x = 10x \]

Step 3: Match Simplified Expression to Left

Now find which expression on the left corresponds to the simplified expression \(10x\): - (c) \(10x\) directly matches with \(10x\).

Key Concepts

Simplifying Expressions

When you see a mathematical expression, it may look a bit complex or jumbled up. The idea behind simplifying expressions is to take this complexity and make it as straightforward as possible. But what does it mean to simplify? It means rewriting the expression in its most basic form without changing its value. This process allows you to work with expressions more easily and efficiently.

For instance, if you're given an expression like \(5x + 5x\), you can simplify it by recognizing that both terms contain the variable \(x\). Here, instead of dealing with two separate parts, you can combine them into a single entity:

• Add the coefficients of \(x\) (which are 5 and 5) to get 10.
• The expression then simplifies to \(10x\).

Simplifying expressions helps in reducing errors, especially in larger equations, and makes subsequent mathematical tasks way less daunting.

Combining Like Terms

A key concept in simplifying expressions is combining like terms. Like terms are terms that contain the exact same variables raised to the same powers, but potentially different coefficients. This principle is fundamental in algebra as it allows mathematical expressions to be tidied up and presented in a clearer form.

To combine like terms, consider the example of the expression \(5x + 5x\) again. Both terms are like terms because they have the same variable, \(x\), raised to the same power, which is 1 in this case. You combine them by adding their coefficients:

• The coefficient for each \(x\) is 5.
• Adding these coefficients (5 + 5) gives you 10, resulting in the simplified expression \(10x\).

This process not only simplifies equations but also sets the foundation for more advanced algebraic operations, making problems easier to solve and understand.

Polynomial Expressions

Polynomial expressions are a type of algebraic expression that involve variables and coefficients, with operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Examples include \(9y - 6y^2\) and \(25x^2\). Being familiar with them is crucial as they form the backbone of many algebraic operations.

Polynomials come in different "degrees," determined by the highest power of the variable present. For example:

• \(9y - 6y^2\) is a quadratic polynomial because the highest power of \(y\) is 2.
• \(25x^2\) is a monomial (a polynomial with only one term), and its degree is 2.

Understanding polynomial expressions is essential for operations like addition, subtraction, or finding roots. Simplifying these expressions, like any other, often involves combining like terms and keeping an eye on the degrees to ensure proper simplification.