Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$\frac{x}{3}-5 x^{2}+x^{3}$$

Step-by-Step Solution

Answer

The terms are \(\frac{x}{3}\), \(-5 x^{2}\), and \(x^{3}\). The coefficients of these variable terms are \(1/3\), \(-5\), and \(1\) respectively.

Step 1 - Identify the terms

The given expression is: \( \frac{x}{3}-5 x^{2}+x^{3} \) our first task is to identify the terms in this expression. Three things are separated by minus and plus signs. So we can say that this expression contains three terms. These are: \(\frac{x}{3}\), \(-5 x^{2}\), and \(x^{3}\)

Step 2 - Identify variable terms

Variable terms are terms that include a variable. In our case, all of our terms are variable terms since each one includes the variable 'x'.

Step 3 - Identify and list the coefficients of each term

The coefficient of a term is the number that is multiplying the variable. Be aware that if a term is just a variable without a numerical multiplier, the coefficient is 1. So, in this case, the coefficients for each term are: For \(\frac{x}{3}, 1/3\), for \(-5 x^{2}, -5\), and for \(x^{3}, 1\)

Key Concepts

Coefficient

In the world of polynomials, a coefficient is like the partner that stands alongside a variable. Think of it as the multiplier of the variable part of each term in an algebraic expression. For example, in the term \(-5x^{2}\), the coefficient is \(-5\). Coefficients can be positive or negative numbers, and they play a crucial role in defining the term's value and impact.

• If a term is simply a variable like \(x\), its implied coefficient is \(1\), even though it isn't written.
• Fractions can also serve as coefficients, such as \(1/3\) in the term \(\frac{x}{3}\).

Understanding coefficients is key when simplifying or evaluating expressions, as they determine how much each term contributes to the overall expression.

Variable Terms

Variable terms are any part of an algebraic expression that includes a variable. In simple terms, they contain letters like \(x\) or \(y\) that stand for numbers we don't yet know. In our given expression \(\frac{x}{3} - 5x^{2} + x^{3}\), all three parts are variable terms because they all have the variable \(x\).Variable terms can tell you a lot about an expression:

• The degree of the variable (such as squared or cubed) indicates the term's impact on the graph of the polynomial.
• The variable terms can hint at how complex solving an equation might be.

Recognizing these terms helps you understand what variables are involved and how they interact in algebraic equations.

Algebraic Expressions

An algebraic expression is essentially a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In the expression \(\frac{x}{3} - 5x^{2} + x^{3}\), we combine all these elements to form a mini mathematical formula. This allows us to represent real-world situations using unknown values.Key aspects of algebraic expressions:

• They do not include equality signs \((=)\), so they're different from equations.
• Expressions can be simple, like \(3x\), or complex, like the one provided.
• The structure of an expression matters. The order and combination of terms can change its meaning and results.

Algebraic expressions form the backbone of most algebra problems, allowing us to model problems and scenarios systematically.