Problem 31 - Algebra A Combined Function

Question

Simplify each expression by combining like terms. See Examples 6 through 10. $$ 7 x^{2}+3+9 x^{2}-10 $$

Step-by-Step Solution

Answer

The simplified expression is \(16x^2 - 7\).

Step 1: Identify Like Terms

First, identify the terms in the expression that have the same variable raised to the same power. In this expression, we have the quadratic terms \(7x^2\) and \(9x^2\), and the constant terms \(+3\) and \(-10\).

Step 2: Combine Like Terms

Next, combine the like terms by adding their coefficients. Start with the quadratic terms: \(7x^2 + 9x^2 = 16x^2\). Then, add the constant terms: \(+3 - 10 = -7\).

Step 3: Write the Simplified Expression

Now, write down the simplified expression by combining the results from the previous step: \(16x^2 - 7\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is one of the foundational skills for simplifying expressions. When you have expressions with the same variable part—meaning they have the same variables raised to the same powers—they are considered "like terms." For example, in the expression given, both \(7x^2\) and \(9x^2\) are like terms because they share the same variable \(x\) raised to the power of two.

• Identify: Look for terms with the same variables raised to identical powers.
• Combine: Sum the coefficients of these like terms to simplify the expression.

In our example, we identified \(7x^2\) and \(9x^2\) as like terms. By adding the coefficients (7 and 9), we combine them to get \(16x^2\). Likewise, we do the same for constant terms like \(+3\) and \(-10\), yielding \(-7\). After combining, the simplified expression becomes \(16x^2 - 7\).
This process helps in making the expressions easier to work with, especially when solving equations.

Quadratic Expressions

Quadratic expressions are a special kind of algebraic expression where the highest power of the variable is two. These expressions are fundamental in algebra and appear often in various problems. The standard form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

In the current exercise, we focus on the first part of the quadratic form, \(ax^2\), where \(a\) represents the coefficient. The expression \(7x^2 + 9x^2\) involves terms where the variable part \(x^2\) is squared. By combining these terms through addition, they simplify into a single quadratic term, \(16x^2\), which is much easier to handle. Recognizing and simplifying such expressions is crucial when dealing with equations like quadratic equations, curve sketching, and optimization problems in mathematics.

Coefficients

Coefficients in algebra are the numerical factors that multiply a variable within a term. They are essential in determining how much of the variable you have. In any term involving a variable, the coefficient is the number directly in front of the variable.

• In \(7x^2\), the number 7 is the coefficient.
• In \(9x^2\), the number 9 is the coefficient.

Coefficients allow us to combine like terms. For instance, by adding the coefficients of \(7x^2\) and \(9x^2\), we consolidate them into a single term, \(16x^2\). This not only simplifies the expression but also makes it easier to evaluate or substitute values. Understanding coefficients is crucial for performing operations like distribution, factoring, and solving equations.

Constant Terms

Constant terms are the portions of an algebraic expression that do not involve any variables. They are simply numbers that stand alone, and they remain unchanged, hence the name "constant." In any expression, identifying and working with these terms is key to simplification.

For example, in the expression \(+3\) and \(-10\) are constant terms. When simplifying, we combine these constants to keep the expression tidy: \(+3 - 10 = -7\).

• They do not change with the variable's changes.
• They determine the vertical shift of a graph if the expression represents a function.

Constant terms are significant when you're solving for specific values, integrating expressions, or understanding overall shifts in function behavior.