Problem 44 - Algebra 1: Concepts and Skills

Question

Use a horizontal format to add or subtract. $$ \left(3 a^{3}-4 a^{2}+3\right)-\left(a^{3}+3 a^{2}-a-4\right) $$

Step-by-Step Solution

Answer

The simplified form of the given polynomials is \(2 a^3 - 7 a^2 + a + 7\)

Step 1: Rearrange the equation

Write the equation without the brackets, making note to change the signs as we are subtracting the second polynomial from the first: \(3 a^{3}-4 a^{2}+3 - a^{3} - 3 a^{2} + a + 4\)

Step 2: Combine like terms

To simplify the equation, look for like terms and combine them. Like terms are those that have the same variable and exponent. The like terms here are \(3a^3\) and \(-a^3\) for the cubic power of \(a\), \(-4a^2\) and \(-3a^2\) for the square of \(a\) terms, \(a\) for the first-power of \(a\) terms, and \(3+4\) for the constant terms. Combine these like terms to simplify: \( (3a^3 - a^3) + (- 4a^2 - 3a^2) + a + (3+4) \)

Step 3: Perform the operations

Perform the addition and subtraction: \(2 a^3 - 7 a^2 + a + 7\)

Key Concepts

Polynomial Addition

Polynomial addition (and subtraction) is a process of combining two or more polynomials into another polynomial. It's an essential skill in algebra, which involves adjusting the signs when the operation involves subtraction, as was the case in the exercise above. When adding polynomials, you simply need to add the coefficients of terms that are alike.

• Write down each polynomial expression clearly with each term lined up.
• For subtraction, remember to distribute the negative sign across the terms of the second polynomial.
• Once all terms are written out, focus on each degree (powers of variables) individually to add or subtract.

A big chunk of this method relies on understanding how to manage these different terms, and that's where like terms come into play.

Like Terms

Like terms are crucial in polynomials, as they share the same variables raised to the same powers. In simpler terms, they are terms that "look alike." For example, in the given exercise, terms like \(3a^3\) and \(-a^3\) are considered like terms because they both have the variable \(a\) raised to the third power.

• Identify terms with identical variables and exponents.
• Pay attention to the coefficients, which can be positive or negative.

Recognizing like terms allows us to combine them easily, thus simplifying the polynomial expression. This skill is particularly helpful when performing operations such as polynomial addition or subtraction.

Combining Like Terms

Combining like terms is the method of simplifying polynomials by adding or subtracting coefficients of like terms. This process helps in reducing the polynomial to its simplest form.

• First, identify all the like terms in the polynomial expression.
• Combine the coefficients of these like terms together.
• Write down the simplified expression after combining like terms.

In our exercise, after identifying like terms, we combined them: \(3a^3 - a^3\) leads to \(2a^3\), and so on, resulting in a neater and more concise expression: \(2a^3 - 7a^2 + a + 7\). This simplification makes it easier to understand the behavior of the polynomial.

Horizontal Format

The horizontal format of adding or subtracting polynomials is a method where the expressions are aligned horizontally, and similar terms are directly combined. This approach is less about vertical alignment but rather clarity when handling each term.

• Begin writing your expressions clearly side by side.
• Ensure the operation (addition or subtraction) between polynomials is correctly applied.
• Read across the line to easily identify and combine like terms.

This format helps visually keep track of which terms are being combined, making it less likely to make errors associated with misalignment or sign mistakes. For students new to polynomials, this visual clarity can be particularly beneficial during practice.