Which expression is equivalent to (3x^2-7)

Equivalent Expression

To determine which expression from the given options is equivalent to:

We need to solve each of the provided expressions:

Option 1

Calculate the result:

Option 2

Calculate the result:

Option 3

Calculate the result:

Option 4

Calculate the result:

Thus, the equivalent expression to $(3x^2 - 7)$ is given by Option 3

Concept Explanation

In mathematics, polynomial subtraction refers to the process of subtracting one polynomial from another. Polynomials are algebraic expressions that consist of variables and coefficients, structured in a sum of terms, each termed as a “monomial”. Here is the basic process:

Step-by-Step Procedure

1. Align the Polynomials: Write down the polynomials in a vertical format, aligning like terms i.e., terms with the same power of the variable.
2. Distribute the Negative Sign: If necessary, distribute the negative sign across the terms of the polynomial being subtracted.
3. Subtract Coefficients: Subtract the coefficients of the aligned terms.
4. Combine the Results: Write down the resulting polynomial after performing the subtraction.

Example

Let's illustrate polynomial subtraction with an example. Suppose we want to subtract the polynomial $4x^3 + 3x^2 - 2x + 7$ from $5x^3 - 2x^2 + x - 4$.

1. Align the Polynomials:

2. Distribute the Negative Sign:

3. Subtract Coefficients:

4. Combine the Results:

Important Points

• Like Terms: Make sure to align and combine only the like terms (terms with the same exponent).
• Negative Sign Distribution: Always distribute the negative sign when subtracting polynomials.
• Resulting Polynomial: The resulting polynomial is formed by subtracting the coefficients of the like terms.

Understanding the polynomial subtraction concept helps in simplifying expressions and solving polynomial equations efficiently.

Introduction

Combining like terms is a fundamental process in algebra that simplifies expressions. Like terms are terms that have the same variables raised to the same power, even if they have different coefficients.

Identifying Like Terms

To combine like terms, you first identify the terms that have the same variables and exponents. For example, in the expression:

The like terms are:

• $3x^2$ and $-x^2$ (both have the variable $x$ raised to the power of 2)
• $5x$ and $-2x$ (both have the variable $x$ raised to the power of 1)

Combining the Terms

1. Group the like terms together.
2. Add or subtract their coefficients.

For the expression given above, we can group and then combine like terms:

Combining the coefficients of the like terms, we get:

So, the simplified expression is:

Special Cases

Constants, or terms without variables, are also considered like terms with each other. For instance, in the expression:

The constants $7$ and $5$ can be combined:

Resulting in the simplified form:

Benefits

Combining like terms:

• Simplifies expressions, making them easier to work with.
• Reduces complexity, especially useful in solving equations or inequalities.

Example

Let's look at another example:

First, group the like terms:

Now, combine the coefficients:

The simplified expression is:

Being proficient at combining like terms is essential for simplifying algebraic expressions and solving algebra problems effectively.