Problem 41 - Algebra 1: Concepts and Skills

Question

Use a horizontal format to add or subtract. $$ \left(z^{3}+z^{2}+1\right)-z^{2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( z^{3} + 1 \).

Step 1: Rewriting the expression

The given expression can be rewritten as \(z^{3}+z^{2}+1+ -z^{2}\). Here, the subtraction operation has been written as the addition of the negative to follow the rules of subtraction in algebra.

Step 2: Grouping

Let's group together the like terms. We can rewrite the expression as \( z^{3} + (z^{2} - z^{2}) + 1\).

Step 3: Simplify

Now, simplifying the expression obtained in the previous step we get \( z^{3} + 0 + 1 \). So the simplified form will be \( z^{3} + 1 \).

Key Concepts

Subtraction of Polynomials

Working with polynomials often involves operations like addition and subtraction, similar to how we handle numbers. However, with polynomials, we must deal with terms of different degrees or powers. The subtraction of polynomials requires:

• Changing the subtraction problem into an addition problem by adding the inverse of the second polynomial. For example, in our exercise, we have \( (z^{3} + z^{2} + 1) - z^{2} \). This can be rewritten as \( z^{3} + z^{2} + 1 + (-z^{2}) \).
• This step ensures that we follow an important algebraic principle: subtraction can be expressed as the addition of a negative.

To subtract polynomials accurately, always apply this idea as your starting point. It's a surefire way to prevent any mix-ups with signs.

Simplifying Expressions

Simplification in polynomials is all about reducing an expression to its simplest form. After rewriting our expression as addition, the next step is to simplify. This involves:

• Combining constants or terms that can be summed without changing the expression's value.
• It often means performing arithmetic and canceling out terms that add up to zero, which is exactly what we've done in our example by grouping \( z^{2} - z^{2} \).

This results in zero: \( (z^{2} - z^{2}) \to 0 \). After simplification in the exercise, the expression boiled down to something much simpler: \( z^{3} + 0 + 1 \), which is simply \( z^{3} + 1 \).
Simplifying makes your polynomials easier to work with in further calculations, so it is a crucial skill to develop!

Combining Like Terms

To simplify expressions efficiently, especially when working with polynomials, the ability to combine like terms is essential. Terms are considered "like" if they have the same variable raised to the same power.

• Identify terms that share the same variable and power, such as \( z^{2} \) terms in the example problem.
• Once you identify like terms, add or subtract the coefficients. In the exercise, \( z^{2} - z^{2} \) results in zero because the coefficients (1 and -1) cancel out.

By effectively combining like terms, you're able to tidy up the polynomial into its most user-friendly form. This process not only simplifies the expression but also lays the groundwork for further solving or factoring. Whether you're adding or subtracting polynomials, your ability to spot and combine like terms will streamline your calculations.