91Ó°ÊÓ

In algebra, **like terms** are terms that have the same variable raised to the same power. For instance, in the expression \((3x^2 + 2x + 1) + (2x^2 - 7x + 5)\), you can spot like terms by comparing the exponents of \(x\). The like terms here are:

• \(3x^2\) and \(2x^2\) since both have \(x^2\)
• \(2x\) and \(-7x\) since both have \(x^1\)
• Constant terms: \(1\) and \(5\)

Recognizing like terms is crucial for simplifying algebraic expressions, as only like terms can be combined. Always ensure you are matching terms that share both the same variable and the same exponent before proceeding to combine them. This is a fundamental concept that underpins solving, simplifying, and understanding polynomial equations.

Once like terms are identified, **combining terms** becomes the next logical step. This involves adding or subtracting the coefficients—the numerical parts of the terms—while leaving the variable and its exponent unchanged. Let's take a closer look at the example:

• Combine \(3x^2\) and \(2x^2\) by adding their coefficients: \(3 + 2 = 5\), resulting in \(5x^2\)
• For \(2x\) and \(-7x\), the coefficients \(2\) and \(-7\) are added to get \(-5x\)
• For the constant terms \(1\) and \(5\), simply add them to yield \(6\)

So, the combining of terms gives you \(5x^2 - 5x + 6\). This simplification helps in reducing the complexity of expressions and makes solving equations easier. It's essential to pay attention to the sign (positive or negative) of the coefficients during this process, as it affects the resulting polynomial.

**Algebraic expressions** are a combination of numbers, variables, and arithmetic operations. They form the backbone of algebra and can range from simple expressions like \(x + 2\) to intricate polynomials like \(3x^2 + 2x + 1\). The expression we are working with, \((3x^2 + 2x + 1) + (2x^2 - 7x + 5)\), is a sum of two polynomials. Key features of algebraic expressions include:

• **Terms**: Each part of the expression, separated by a '+' or '-', is a term.
• **Coefficients**: The numerical part of a term. For example, in \(3x^2\), \(3\) is the coefficient.
• **Variables**: Symbols that represent numbers, such as \(x\).
• **Exponents**: Indicate how many times to multiply the variable by itself, such as \(x^2\).

Understanding how to manipulate these expressions is foundational in algebra. It enables students to simplify equations, solve for variables, and understand complex relationships in mathematics. Simplifying expressions by combining like terms is just one of the many skills learned when handling algebraic expressions.