Problem 21 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ 2(2-5 t)+t^{2}(t-1)-\left(t^{4}-1\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-t^4 + t^3 - t^2 - 10t + 5\).

Step 1: Distribute the constants

Start by distributing the constants in the expression. There are two places where distribution is needed: for the term \(2(2-5t)\), distribute 2 to both terms inside the parentheses.\[2 \times 2 - 2 \times 5t = 4 - 10t\]The rest of the expression currently remains the same:\[t^2(t-1) - (t^4 - 1)\]So, the expression becomes:\[4 - 10t + t^2(t-1) - (t^4 - 1)\]

Step 2: Expand the polynomial expressions

Next, expand the polynomial expressions where necessary. Specifically, for the term \(t^2(t-1)\), multiply \(t^2\) by each term within the parenthesis.\[t^2 \cdot t - t^2 \cdot 1 = t^3 - t^2\]Now substitute this back into the expression:\[4 - 10t + t^3 - t^2 - (t^4 - 1)\]

Step 3: Remove the grouping symbol

In this step, remove the parentheses with the negative sign in front of \((t^4 - 1)\). Distribute the negative sign across the terms inside.\[-(t^4 - 1) = -t^4 + 1\]So the expression becomes:\[4 - 10t + t^3 - t^2 - t^4 + 1\]

Step 4: Combine like terms

Now, we need to combine the like terms to simplify the expression. Gather all terms with the same power of \(t\):\[\- t^4 + t^3 - t^2 - 10t + 4 + 1\]Combine the constant terms and arrange terms by descending powers of \(t\):\[\- t^4 + t^3 - t^2 - 10t + (4 + 1)\]Simplify the constants:\[\- t^4 + t^3 - t^2 - 10t + 5\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term across terms inside a set of parentheses. This principle is helpful for breaking down complex expressions into manageable pieces.
This property states that for any numbers or variables, and any expression inside parentheses, you can distribute the multiplication over addition or subtraction. In general form, it is shown as:

• \(a(b + c) = ab + ac\)

In our original exercise, we applied the distributive property to the expression \(2(2-5t)\). Here, 2 is multiplied by each term inside the parentheses. We solved it as follows:

• Multiply 2 by 2, giving us \(4\).
• Then, multiply 2 by \(-5t\), resulting in \(-10t\).

After applying the distributive property, the expression is transformed into \(4 - 10t\). Understanding this property is key to mastering expressions that involve multiplication across parentheses.

Combining Like Terms

Combining like terms is an essential skill in algebra for simplifying expressions by adding or subtracting terms that have the same variables raised to the same power.
This allows for a more concise and manageable form of the expression.
Like terms are terms that contain the same variable or variables with the same exponent. To combine them, you simply add or subtract their coefficients.

• For example, in the expression \(-t^4 + t^3 - t^2 - 10t + 4 + 1\), you look for terms with identical variable parts.
• Here, \(4\) and \(1\) are like terms as they are constants, which can be combined as \(4 + 1\ = 5\).

Once we have found all like terms, we rewrite the expression with combined coefficients. Thus, the expression \(-t^4 + t^3 - t^2 - 10t + 5\) is obtained. Recognizing and combining like terms is crucial for finding the simplest form of a polynomial.

Polynomial Expansion

Polynomial expansion involves multiplying terms in a polynomial and simplifying into a longer expression where terms might need to be combined. This is a key step in simplifying expressions that contain parentheses.
In our exercise, we expanded \(t^2(t-1)\). We did this by multiplying each term inside the parentheses by \(t^2\), as follows:

• Multiply \(t^2\) by \(t\), which results in \(t^3\).
• Multiply \(t^2\) by \(-1\), yielding \(-t^2\).

This transforms \(t^2(t-1)\) into \(t^3 - t^2\).
By expanding the parentheses, we split complex terms into simpler parts open for simplifying. Polynomial expansion often involves both applying the distributive property and then combining like terms.