Problem 30 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (4 x-1)(3 x+7) $$

Step-by-Step Solution

Answer

The simplified expression is \(12x^2 + 25x - 7\).

Step 1: Apply the Distributive Property

Use the distributive property to expand the expression. Multiply each term in the first bracket by each term in the second bracket:\[ (4x - 1)(3x + 7) = 4x(3x) + 4x(7) - 1(3x) - 1(7) \]

Step 2: Multiply Each Pair of Terms

Perform each multiplication individually:- \(4x \times 3x = 12x^2\)- \(4x \times 7 = 28x\)- \(-1 \times 3x = -3x\)- \(-1 \times 7 = -7\)Combine these results:\[ 12x^2 + 28x - 3x - 7 \]

Step 3: Combine Like Terms

Add or subtract like terms (terms with the same variable and exponent) in the expression:- Combine \(28x - 3x = 25x\)Now the expression becomes:\[ 12x^2 + 25x - 7 \]

Key Concepts

Distributive Property

The distributive property is fundamental in algebra and aids in expanding expressions effectively. When dealing with expressions like \((a + b)(c + d)\), the distributive property allows us to rewrite this as:

• \(a(c + d) + b(c + d)\)
• which then becomes \(ac + ad + bc + bd\)

This property tells us to distribute each term in the first pair of parentheses to every term in the second pair.

In the exercise provided, \((4x - 1)(3x + 7)\), the distributive property helps by breaking down the expression into individual multiplications:

• \(4x\) is distributed across \((3x + 7)\), producing \(4x(3x) + 4x(7)\).
• Similarly, \(-1\) is distributed, giving \(-1(3x) - 1(7)\).

Understanding how to apply the distributive property is key to manipulating expressions and solving equations efficiently.

Combining Like Terms

After distributing terms in an expression, the next crucial step is to combine like terms to simplify further. "Like terms" refer to terms in an algebraic expression that have the same variables raised to the same power.

In the expression derived from our exercise, \(12x^2 + 28x - 3x - 7\), we notice the terms \(28x\) and \(-3x\) are like terms. Both have the variable \(x\), raised to the first power.

• Adding them together simplifies to: \(28x - 3x = 25x\).

Combining like terms simplifies the expression, making it more manageable and easier to understand or solve. This step is crucial as it reduces complexity in algebraic expressions, ensuring clarity and accuracy.

Simplification of Expressions

Simplifying expressions makes them easier to work with, and it involves a systematic reduction of complexity in mathematical statements. Once we've distributed and combined like terms, our focus shifts to ensuring the expression is in its simplest form.

From our worked example, after applying the distributive property and combining like terms, we arrived at \(12x^2 + 25x - 7\). This expression is now simplified, meaning no further reduction is possible without changing its value.

• The coefficients are neatly combined.
• There are no parentheses or unnecessary variables to carry further.
• All similar terms have been combined, offering a clear expression for any further calculations or evaluations.

Simplifying expressions is not just procedural; it aids in gaining a deeper understanding as unnecessary elements are removed, revealing the core mathematical relationships.