Problem 86 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 10 y^{2}-8 y+y-7 $$

Step-by-Step Solution

Answer

The simplified expression is \(10y^2 - 7y - 7\).

Step 1: Identify Like Terms

Look at the expression \(10y^2 - 8y + y - 7\). Like terms are those that have the same variable raised to the same power. Here, \(y^2\) terms, \(y\) terms, and constant terms need to be combined separately.

Step 2: Combine \(y\) Terms

The terms \(-8y + y\) both contain \(y\) raised to the power of one. Combine these by adding their coefficients: \(-8 + 1 = -7\). This gives us \(-7y\).

Step 3: Compose the Simplified Expression

After combining like terms, write down the expression again: \(10y^2 - 7y - 7\). This is simplified because all like terms have been combined, and no further simplification is possible.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, a crucial step is combining like terms. Like terms are terms in an expression that have the same variables with the same exponents. For example, in the expression \(10y^2 - 8y + y - 7\), the terms \(-8y\) and \(y\) are like terms because they both involve the variable \(y\) to the power of one. Similarly, constant terms such as \(-7\) are also considered like terms because they do not have any variables.To combine like terms:

• Identify all terms with the same variables and powers.
• Add their coefficients together while keeping the variable part unchanged.

In our example, when combining \(-8y\) and \(y\), you add their coefficients: \(-8 + 1 = -7\), resulting in the new term \(-7y\). The rest of the expression remains unchanged, as there are no other like terms to combine with \(10y^2\) or \(-7\).

Polynomials

Polynomials are algebraic expressions made up of terms that are grouped together. Each term is a product of a constant coefficient and one or more variables raised to whole number exponents. In the expression \(10y^2 - 8y + y - 7\), we see a polynomial with three types of terms:

• The term \(10y^2\) is a quadratic term because its variable, \(y\), is raised to the power of two.
• The terms \(-8y\) and \(y\) are linear terms as their variable \(y\) is to the power of one.
• The term \(-7\) is a constant, as it has no variables.

Polynomials can be classified by their degree, which is determined by the highest power of the variable in the expression. The degree of this polynomial is 2, given by the highest exponent in \(10y^2\). Understanding polynomials and identifying their terms help in organizing and simplifying expressions efficiently.

Simplification

The goal of simplification in algebra is to make expressions easier to work with by reducing them to their most concise form while retaining the same value. Simplifying involves combining like terms, removing parentheses, and performing any calculations that can simplify the structure of the polynomial.The process of simplification provides:

• A clearer and more understandable expression.
• Easier computations for subsequent operations, like solving equations.

In the expression \(10y^2 - 8y + y - 7\), simplification involves combining the like terms \(-8y\) and \(y\) to produce a more streamlined expression: \(10y^2 - 7y - 7\). There are no parentheses to remove and no further operations possible here, indicating the expression is fully simplified. Simplifying expressions is essential in algebra, providing clarity and reducing errors in more complex calculations.