Problem 46 - Algebra 1: Concepts and Skills

Question

Use a vertical format or a horizontal format to add or subtract. $$ \left(-2 t^{4}+6 t^{2}+5\right)-\left(-2 t^{4}+5 t^{2}+1\right) $$

Step-by-Step Solution

Answer

The simplified expression of the original polynomial is \( t^2 + 4 \)

Step 1: Distribute the Subtraction Sign

Firstly, distribute the subtraction sign across each term within the second parentheses, such that we rewrite the expression as \( -2t^4 + 6t^2 + 5 + 2t^4 - 5t^2 - 1 \). It's important to notice that the sign of the terms in the second polynomial all changed.

Step 2: Combine Like Terms

Next, we should combine the like terms. Same powers of \( t \) are like terms. So, we rewrite the expression as \( (-2t^4 + 2t^4) + (6t^2 - 5t^2) + (5 - 1) \). This combines all like terms adjacent to each other for clarity.

Step 3: Simplify the Expression

Finally, we simplify each group of terms, such that the final expression will be \( 0t^4 + t^2 + 4 \) or simply \( t^2 + 4 \) since \( 0t^4 \) effectively disappears from the expression.

Key Concepts

Combining Like Terms

When working with polynomials, one of the most essential skills is combining like terms. Like terms are those that have the same variable raised to the same power. This means if we have terms like \(3t^2\) and \(-5t^2\), they are considered like terms because they both are terms with the variable \(t\) raised to the second power.

To combine these terms, you simply add or subtract their coefficients, while keeping the variable part the same. Let's look at an example:
- Suppose you have the expression \(6t^2 - 5t^2\).
- Here, \(6t^2\) and \(-5t^2\) are like terms, so you subtract 5 from 6, which gives you \(1t^2\).
- The simplified form is \(t^2\) since 1 is the coefficient and can be left out.

In practice, when you're simplifying a polynomial, it's often helpful to first organize all like terms together, especially in more complex expressions. This makes the process of combining them straightforward and less error-prone.

Distributive Property

The distributive property is a fundamental algebraic property that lets us multiply a single term by multiple terms inside a set of parentheses. It’s usually expressed as \(a(b + c) = ab + ac\). This property is particularly handy when dealing with subtraction in polynomials, as it allows us to "distribute" the minus sign to each term inside the parentheses.

In the exercise we are discussing, the expression \(-(-2t^4 + 5t^2 + 1)\) involves distributing the subtraction sign. This results in switching the sign of each term inside the parentheses. Here's how it works step-by-step:

• The negative sign in front of the parentheses changes \(-2t^4\) to \(+2t^4\).
• The \(+5t^2\) becomes \(-5t^2\).
• Finally, \(+1\) turns into \(-1\).

Being comfortable with distributing in this way lays the groundwork for accurately simplifying expressions, as it ensures that all terms are correctly accounted for prior to combining like terms.

Simplifying Expressions

Simplifying expressions is all about making a complicated expression easier to read and work with by combining terms and following algebraic processes to reduce its complexity. The goal is to present the expression in its simplest form.

In our polynomial expression, after distributing and combining like terms, simplifying becomes the final polish to our answer. Let's break down the process:

• After applying distribution and combining like terms, you might get something like \(0t^4 + t^2 + 4\).
• The term \(0t^4\) has a coefficient of zero, which means it won't affect the value of the expression and should be removed.
• You're left with \(t^2 + 4\), where all possible like terms have been combined, and further simplification isn't possible.

This resulting expression is much easier to interpret and often how one would present the solution. By simplifying expressions, we ensure they are in the cleanest and most understandable form, which aids further calculations or interpretations.