Problem 72 - Algebra 1: Concepts and Skills

Question

Add or subtract the polynomials. (Lesson 10.1) $$\left(4 t^{2}+5 t+2\right)-\left(t^{2}-3 t-8\right)$$

Step-by-Step Solution

Answer

The result of the polynomial addition/subtraction is \(3t^2 + 8t + 10\).

Step 1: Distribute the negative sign

The first step is to distribute the negative sign from the second polynomial to each term inside the parentheses. This turns the subtraction problem into an addition problem: \(4t^2 + 5t + 2 - t^2 + 3t + 8\)

Step 2: Combine like terms

Next, rearrange the polynomial so the like terms are together, and then combine them. This results in: \( (4t^2 - t^2) + (5t + 3t) + (2 + 8)\)

Step 3: Simplify

Finally, simplify the polynomial by adding the coefficients of the like terms. This results in: \(3t^2 + 8t + 10\)

Key Concepts

Distributing the Negative Sign

When subtracting polynomials, a common pitfall is omitting the distribution of the negative sign across the terms of the polynomial being subtracted. It's crucial to apply this step before attempting to combine like terms. To distribute the negative sign, simply change the sign of each term inside the parentheses that you're subtracting.

For instance, consider the polynomial subtraction \( (4t^2 + 5t + 2) - (t^2 - 3t - 8) \). The negative sign outside the second set of parentheses must be distributed to each term within: \( -t^2 + 3t + 8 \). This alteration transforms the subtraction into an addition problem, allowing for easier consolidation of like terms in the next step. Without correctly distributing the negative sign, the solution will be inaccurate, which underscores the importance of this preliminary step.

Combining Like Terms

Once the negative sign has been distributed, the next step in simplifying polynomial expressions is to combine like terms. Like terms are terms that have the same variables raised to the same powers. The process involves reordering and grouping these terms to simplify the expression.

Take the distributed equation \( 4t^2 + 5t + 2 - t^2 + 3t + 8 \). To combine like terms, group the terms containing \( t^2 \), those with \( t \), and the constant terms separately: \( (4t^2 - t^2) + (5t + 3t) + (2 + 8) \). Then, you add the coefficients of each group to consolidate the terms. Remember, only the coefficients are added together—the variables and their exponents remain the same. This step is essential for reducing the expression to its simplest form.

Simplifying Polynomials

The final landmark in polynomial subtraction is simplifying the polynomial by adding or subtracting the coefficients of like terms. After combining like terms, each type of term will be represented just once in the simplified expression, making it easier to interpret and use.

In our example, after combining like terms, we get \( (4t^2 - t^2) + (5t + 3t) + (2 + 8) \). Simplifying further, we add the coefficients: \( 3t^2 + 8t + 10 \). There are no more like terms to combine, which means we have completed the process. This simplified polynomial represents the same quantity as the original expression but in a form that is easier to understand and apply in further calculations or when graphing the function. The ability to simplify polynomials is not only useful in basic algebra but also crucial in higher-level math and various applied sciences.