Problem 47 - Elementary and Intermediate Algebra

Question

Perform the operations. See Example 4. $$ (3 x+4)(2 x-2)-(2 x+1)(x+3) $$

Step-by-Step Solution

Answer

The result is \(4x^2 - 5x - 11\).

Step 1: Expand the First Expression

Let's begin by expanding the expression \((3x + 4)(2x - 2)\). We do this using the distributive property (also known as the FOIL method for binomials).First, multiply the first terms: \(3x \cdot 2x = 6x^2\).Next, multiply the outside terms: \(3x \cdot (-2) = -6x\).Then, multiply the inside terms: \(4 \cdot 2x = 8x\).Finally, multiply the last terms: \(4 \cdot (-2) = -8\).Combine these to get: \(6x^2 - 6x + 8x - 8\), which simplifies to \(6x^2 + 2x - 8\).

Step 2: Expand the Second Expression

Next, we need to expand the expression \((2x + 1)(x + 3)\).First, multiply the first terms: \(2x \cdot x = 2x^2\).Next, multiply the outside terms: \(2x \cdot 3 = 6x\).Then, multiply the inside terms: \(1 \cdot x = x\).Finally, multiply the last terms: \(1 \cdot 3 = 3\).Combine these to get: \(2x^2 + 6x + x + 3\), which simplifies to \(2x^2 + 7x + 3\).

Step 3: Subtract the Second Expansion from the First

Now, subtract the expression we obtained from step 2 from the expression in step 1.Subtract:\((6x^2 + 2x - 8) - (2x^2 + 7x + 3)\).This means we distribute the minus through the second set of parentheses:\(6x^2 + 2x - 8 - 2x^2 - 7x - 3\).Combine like terms:\((6x^2 - 2x^2) + (2x - 7x) + (-8 - 3)\).This simplifies to: \(4x^2 - 5x - 11\).

Step 4: Finalize the Result

After combining the terms, the simplified expression from the subtraction is:\(4x^2 - 5x - 11\).This represents the final result of performing the operations on the original expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term within a set of parentheses. This ensures that each term in the parentheses is accounted for in the multiplication process.
Let's examine how it is applied in our exercise:

• First, consider the expression \( (3x + 4)(2x - 2) \). Here, we distribute each term in the first binomial, \(3x + 4\), across every term in the second binomial, \(2x - 2\).
• You multiply the first terms from each binomial: \(3x \cdot 2x = 6x^2\).
• Then, the outside terms: \(3x \cdot (-2) = -6x\).
• Next, the inside terms: \(4 \cdot 2x = 8x\).
• Finally, the last terms: \(4 \cdot (-2) = -8\).

By following these steps, we end up with the expression \(6x^2 - 6x + 8x - 8\). Combining these results using the distributive property helps in systematically expanding polynomials.

Binomial Expansion

Binomial expansion involves the multiplication of two binomials, resulting in a polynomial. Each term of the first binomial is multiplied by each term of the second, and then combined to form a new expression. This process is crucial in simplifying expressions.
Using our exercise:

• Begin with \( (2x + 1)(x + 3) \), applying a similar strategy as before.
• First terms: \(2x \cdot x = 2x^2\).
• Outside terms: \(2x \cdot 3 = 6x\).
• Inside terms: \(1 \cdot x = x\).
• Last terms: \(1 \cdot 3 = 3\).

This gives us \(2x^2 + 6x + x + 3\). By combining like terms, our expression simplifies to \(2x^2 + 7x + 3\). The method requires careful attention to each multiplication step and the organization of like terms.

Combining Like Terms

Combining like terms is essential when simplifying expressions, as it involves adding or subtracting terms that have the same variable component raised to the same power. This process reduces the complexity of a polynomial.
In our problem, after using the distributive property on both expressions, we simplified them to \(6x^2 + 2x - 8\) and \(2x^2 + 7x + 3\).

• Subtract these to get: \(6x^2 + 2x - 8 - (2x^2 + 7x + 3)\).
• First, distribute the negative sign: \(6x^2 + 2x - 8 - 2x^2 - 7x - 3\).
• Next, combine like terms: \( (6x^2 - 2x^2) + (2x - 7x) + (-8 - 3)\).
• The resulting expression is \(4x^2 - 5x - 11\).

By carefully combining like terms, expressions are simplified into a more manageable form for solving or further analysis.