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Chapter 10: Problem 78

Find the product. $$(2 x-4)\left(\frac{1}{4} x-2\right)$$

Short Answer

Expert verified
The product of \( (2x - 4) \) and \( \left( \frac{1}{4}x - 2 \right) \) is \( \frac{1}{2}x^2 - 6 \).

Step by step solution

01

Apply Distributive Property

First we distribute \( (2x - 4) \) over \( \left( \frac{1}{4}x - 2 \right) \) which can be done by multiplying each term of the first bracket with all terms of the second bracket.
02

Multiply

Now multiply \( (2x) \) with \( \left( \frac{1}{4}x \right) \) and \( (2x) \) with \( -2 \) and then \( -4 \) with \( \left( \frac{1}{4}x \right) \) and \( -4 \) with \( -2 \).
03

Simplify Your Expression

After multiplying, simplify the resulting expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplying Binomials
Multiplying binomials involves using a key algebraic principle called the distributive property. This concept allows you to multiply each term in the first binomial with every term in the second binomial. In the exercise given, we have two binomials: \((2x - 4)\) and \(\left( \frac{1}{4} x - 2 \right)\). To find their product, we apply the distributive property:
  • Take the first term in the first binomial \((2x)\) and multiply it by both terms in the second binomial.
  • Next, take the second term in the first binomial \((-4)\) and multiply it by each term in the second binomial.
This results in four individual products that are then combined to form a new polynomial. The precision of distributing each term properly ensures that all parts of the expression are accounted for in the product.
Simplifying Expressions
Once you have multiplied the terms, you will have a new algebraic expression that needs to be simplified. Simplifying means combining like terms to make the expression clearer and more concise. In our example, after multiplying, you might have terms like \(\frac{1}{2}x^2\), \(-4x\), and constants like 8.
  • First, look for any terms with \(x^2\) and combine them, if necessary.
  • Next, do the same for terms with \(x\).
  • Finally, combine constant terms.
Simplification reduces complexity and is a crucial step to check if you've achieved the most reduced form possible. The aim is to make the equation easier to interpret and solve.
Algebraic Multiplication
Algebraic multiplication isn't just about multiplying numbers, it's about understanding relationships between variables and constants. In the original problem, algebraic multiplication required following specific steps:
  • Apply the distributive property as the initial step to deal with binomials.
  • Pay attention to the signs of the terms to avoid simple errors when multiplying negative numbers.
  • Remember to handle coefficients correctly to ensure accurate multiplication of variable terms.
The result of algebraic multiplication is a polynomial that reveals new terms and sometimes, new solutions. By mastering this process, you'll have comprehensive skills to tackle a variety of algebraic problems efficiently.

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