91Ó°ÊÓ

Simplifying expressions is all about making an algebraic expression as simple as possible. It involves processes such as eliminating unnecessary terms and combining like terms.
By simplifying, we make an expression easier to understand or solve, without changing its value.

• Start with the given expression: Make sure you carefully write down all parts of the expression before you begin simplifying.
• Apply operations carefully: This includes steps like multiplying, adding, or distributing as needed.
• Combine like terms: Search for terms that have the same variables and exponents, and add or subtract them together.
• Check your work: Always re-evaluate your expression to ensure it’s in its simplest form.

Simplifying is like cleaning up, ensuring that your mathematical expressions are neat and easy to understand.

The distributive property is a handy tool that allows you to multiply a single term by each term within a set of parentheses.
For example, in the expression \(a(b + c)\), you distribute \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\).

Example

Let's take \(-3(x^2 + 1)\). The distributive property lets us multiply \(-3\) by \(x^2\) and \(-3\) by \(1\)

• Distribute \(-3 \times x^2\) gives \(-3x^2\).
• Then, \(-3 \times 1\) gives \(-3\).

Now, the part of the expression inside the parentheses \((x^2 + 1)\) is transformed into \(-3x^2 - 3\).
This property is crucial in simplifying expressions because it enables us to effectively "distribute" multiplication across addition or subtraction inside parentheses.

Combining like terms is a method used to simplify algebraic expressions by merging terms with the same variables and exponents.
This helps in reducing the complexity of an expression.

Understanding Like Terms

Like terms have identical variable parts, so only their coefficients can be different.

• For example, in \(+ 3x^2\) and \(-3x^2\), the variable part \(x^2\) is the same.

In a simplified expression, you calculate the sum or the difference of the coefficients of these terms.

Example

Take the expression \(x^4 + 3x^2 - 3x^2 + 2 - 3\).

• Combine \(3x^2\) and \(-3x^2\) to get \(0\). These terms cancel each other out.
• Then, handle the constant terms: \(+2\) and \(-3\), resulting in \(-1\).
• The \(x^4\) term doesn’t have any like terms to combine.

Finally, by combining like terms, the expression simplifies to \(x^4 - 1\). This step was important in obtaining the most compact form of the expression.