Problem 49 - College Algebra

Question

For the following exercises, find the sum or difference. $$(4 t-x)(t-x+1)$$

Step-by-Step Solution

Answer

The expanded expression is \(4t^2 - 5tx + x^2 + 4t - x\).

Step 1: Expand the expression

To expand the expression \((4t - x)(t - x + 1)\), apply the distributive property \((a+b)(c+d) = ac + ad + bc + bd)\), treating it as distributing each term of the second binomial across the terms in the first binomial.

Step 2: Distribute first term of second binomial

First, distribute the \(t\) across the terms in \(4t-x\):\[t(4t - x) = 4t^2 - tx\].

Step 3: Distribute second term of second binomial

Now distribute \(-x\) across the terms in \(4t-x\):\[-x(4t - x) = -4tx + x^2\].

Step 4: Distribute third term of second binomial

Finally, distribute the constant term \(+1\) across the terms in \(4t-x\):\[1(4t - x) = 4t - x\].

Step 5: Combine all distributed terms

Now, collect all the terms obtained from each distribution:\[4t^2 - tx - 4tx + x^2 + 4t - x\].

Step 6: Simplify the expression

Combine like terms in the above expression:1. Combine \(-tx\) and \(-4tx\) to get \(-5tx\).2. Combine all terms to get: \[4t^2 - 5tx + x^2 + 4t - x\].

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra, used to simplify expressions and solve equations. At its core, it helps to distribute a single term across terms inside parentheses. This concept is often written as \[a(b + c) = ab + ac.\]In our exercise, we apply this property to the binomial terms \((4t - x)(t - x + 1)\).
Here’s a simple breakdown:

• Take each term from the first binomial \((4t - x)\).
• Multiply it by each term in the second group \((t - x + 1)\).

This results in multiple individual multiplication operations, which leads to an expanded expression. It's a vital operation because it transforms product formats into sums, making them easier to handle in subsequent steps. Always remember to carefully manage signs (like negative signs) to keep the process accurate.

Expanding Expressions

Expanding expressions like \((4t - x)(t - x + 1)\) means converting products into sums and differences. This process actually involves repeated use of the distributive property.
In our example, we perform the following:

• First multiply \(t\) by each term in \(4t - x\), resulting in \(4t^2 - tx\).
• Then multiply \(-x\) by \(4t - x\) to get \(-4tx + x^2\).
• Finally, multiply \(+1\) by \(4t - x\) to yield \(4t - x\).

This expansion takes us from two binomials to a longer polynomial form, laying the groundwork for combining like terms. The process might seem tedious, but it's essential for problem-solving, as it converts expressions into manageable components.

Combining Like Terms

Once you expand an expression, you'll often find like terms that can be simplified to make the expression easier to work with. Like terms are those that contain the same variables raised to the same power. In our provided example, from the expression \[4t^2 - tx - 4tx + x^2 + 4t - x,\]we identify terms that can be combined:

• Combine \(-tx\) and \(-4tx\), as both have the same variables: resulting in \(-5tx\).
• Notice terms like \(4t^2\), \(x^2\), \(4t\), and \(-x\) each stands separately.

Thus, the expression simplifies to \[4t^2 - 5tx + x^2 + 4t - x.\]Combining like terms reduces clutter and streamlines expressions, a key step in algebra that aids in solving and simplifying equations. Keep an eye out for coefficients and signs as they play a crucial role in properly combining terms.