Problem 13 - Algebra 1

Question

Find the terms of the expression. \(-7 y^{2}+12 y-6\)

Step-by-Step Solution

Answer

The expression \(-7 y^{2}+12 y-6\) is composed of the terms \(-7 y^{2}\), \(12 y\), and \(-6\).

Step 1: Analyze the algebraic expression

In the given expression \(-7 y^{2}+12 y-6\), the terms are the elements that are added or subtracted.

Step 2: Identify the terms

Examining closely, we see that the expression is the sum (or difference) of three terms: \(-7 y^{2}\) is the first term, \(12 y\) is the second term, and \(-6\) is the third term.

Step 3: Present the solution

The expression \(-7 y^{2}+12 y-6\) is composed of the terms \(-7 y^{2}\), \(12 y\), and \(-6\).

Key Concepts

Terms of an Expression

In algebra, an expression is made up of terms. Terms are the individual parts separated by addition or subtraction within an expression, like puzzle pieces completing a picture. For example, in the expression \(-7y^{2} + 12y - 6\), there are three terms. Each term in this expression, namely \(-7y^{2}\), \(12y\), and \(-6\), carries unique attributes: some include variables, others are constants. This separation into terms is crucial for understanding algebraic expressions.

When analyzing expressions, always look for these addition and subtraction signs; they are the separators indicating new terms.

• Terms can include variables (letters that represent numbers).
• They can also be constants (fixed numbers without attached variables).
• During your calculations, treat each term as a standalone unit until required to perform operations involving more than one of them.

Polynomial

A polynomial is a special type of algebraic expression where the terms are powers of a variable. Each term in a polynomial has the form \(a_nx^n\), where \(a_n\) is a coefficient and \(x^n\) is a variable raised to a non-negative integer power. In our expression, \(-7y^{2} + 12y - 6\), we see it fits this description, making it a polynomial.

Polynomials can be categorized by their degree, which is the highest power of the variable in the terms of the expression:

• The degree of \(-7y^{2}+12y-6\) is 2, which indicates a quadratic polynomial because \(-7y^{2}\) is the highest power term.
• Degrees give us insight into the polynomial's behavior, such as the number of solutions it can have.

Understanding the structure of polynomials helps in grasping concepts related to algebraic equations and further applications in calculus and other fields.

Coefficients

Coefficients are numerical factors in terms of algebraic expressions. They're the numbers directly multiplying the variable part in terms. For instance, in the polynomial \(-7y^{2} + 12y - 6\):

• The coefficient of \(y^{2}\) is \(-7\).
• For the term \(12y\), the coefficient is \(12\).
• The term \(-6\) is technically a constant term, with an implicit variable raised to power zero, and a coefficient of \(-6\).

Coefficients play a pivotal role in determining the influence of each term within an expression.

Not only do they define the numerical value of the term, but they also affect how expressions are added, subtracted, and otherwise manipulated within mathematical operations. Remember, in calculus and algebra, changing a coefficient can alter the graph and solutions of an equation significantly.