Problem 8 - Algebra 1: Concepts and Skills

Question

Simplify the expression by combining like terms if possible. If not possible, write already simplified. $$ 8-4 t+6 t^{2} $$

Step-by-Step Solution

Answer

The algebraic expression is already simplified, and is rendered as 8 - 4t + 6t^2.

Step 1: Identifying Like Terms

In the given expression 8 - 4t + 6t^2, examine each term individually. We have three terms: 8, -4t, and 6t^2. The terms -4t and 6t^2 both contain the variable 't', but because the variable is raised to different powers, they cannot be combined. Even though -4t and 6t^2 have the similar variable, they are not considered as like terms.

Step 2: Combining Like Terms

In this case, we don't have any like terms. This indicates we are unable to simplify the expression any further by combining terms.

Step 3: Writing Final Expression

There are no like terms we can combine therefore, our simplified expression remains as 8 - 4t + 6t^2.

Key Concepts

Combining Like Terms

Simplifying algebraic expressions is akin to tidying up a mixed bag of elements by grouping similar ones together. Combining like terms is a fundamental procedure in this process. Like terms are those terms in an expression that have the exact variable raised to the same power. For instance, in an expression such as \( 3x + 2x^2 - 5x \), the like terms are the first and last terms because they both are simply multiples of \( x \) and can be combined.

To simplify the expression, you would add the coefficients of these like terms together, resulting in \( (3 - 5)x + 2x^2 \) which further simplifies to \( -2x + 2x^2 \). Note how we treated the coefficients as regular numbers, leaving the variable part of the term unchanged. It's essential to acknowledge that only the coefficients of like terms can be combined; the variables remain untouched unless the terms are identical in variable composition and power.

Algebraic Terms and Coefficients

Algebraic expressions are constructed from terms stitched together by addition and subtraction. A term is composed of a coefficient and a variable part. The coefficient is a numerical multiplier of the variable part, representing how many units of this variable you have. In the expression \( 8 - 4t + 6t^2 \), the term -4t has a coefficient of -4, and the term \( 6t^2 \) has a coefficient of 6.

Coefficients can be positive or negative, whole numbers, fractions, or decimals. It is crucial to understand that while -4t and \( 6t^2 \) both contain the variable 't', coefficients alone do not determine if terms are 'like terms'. The variable part, including its power, must also be the same. So when scrutinizing an algebraic expression, always consider both the coefficients and the powers of the variables before deciding to combine any terms.

Simplification of Polynomials

When dealing with polynomials, which are algebraic expressions containing multiple terms, simplification entails reducing them to their most concise, efficient form. The step-by-step solution to the exercise \( 8 - 4t + 6t^2 \) points out a key principle in the simplification process: only combine terms with identical variable parts raised to the same exponent.

While the given expression cannot be simplified by combining like terms, it's essential to know that simplification can also involve factoring, cancelling, or expanding expressions, depending on what is being asked. For polynomials, always begin by looking for like terms to combine and continue to factor or expand if necessary. The lack of like terms in this particular expression means it is already as simple as it can be, and your focus then shifts to proper formatting and presentation of the final answer.