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Chapter 5: Problem 55

Products of Sums and Differences Multiply. $$(1-4 x)(1+4 x)$$

Short Answer

Expert verified
The product is \[1 - 16x^2\].

Step by step solution

01

Identify the Type of Expression

Notice that we are multiplying two binomials: \(1 - 4x\) and \(1 + 4x\). This is a product of a sum and a difference.
02

Recall the Formula for Product of Sums and Differences

The formula for the product of a sum and a difference is \[ (a - b)(a + b) = a^2 - b^2 \]. Here, \(a = 1\) and \(b = 4x\).
03

Apply the Formula

Using the formula, substitute the values of \(a\) and \(b\): \[(1 - 4x)(1 + 4x) = 1^2 - (4x)^2\].
04

Calculate Each Term

Calculate \(1^2\) and \((4x)^2\): \[1^2 = 1 \] and \[(4x)^2 = 16x^2\].
05

Simplify the Expression

Subtract the results to get the final simplified expression: \[1 - 16x^2\].

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Key Concepts

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

binomial multiplication
When dealing with binomial multiplication, you multiply two binomials together. A binomial is a polynomial with two terms. For example, in our given problem, we have the binomials \(1 - 4x\) and \(1 + 4x\). To simplify the multiplication, there are methods tailored to different types of binomials, such as the FOIL method or using special formulas. For sums and differences, the special formula \((a - b)(a + b) = a^2 - b^2\) comes in handy. Understanding and identifying the type of binomials simplify the calculation a lot.
difference of squares
The formula for the difference of squares is used when you multiply a sum and a difference of the same terms, like \(a - b\) and \(a + b\). The difference of squares formula states that \((a - b)(a + b) = a^2 - b^2\). This property is very useful because it turns a binomial product into a simpler subtraction of squares. In the problem \(1 - 4x\) and \(1 + 4x\), the value of \(a\) is \1\ and the value of \(b\) is \4x\. So, according to the formula, \((1 - 4x)(1 + 4x) = 1^2 - (4x)^2 \), which simplifies to \1 - 16x^2\. This shows how a product of binomials is simplified into a single expression.
intermediate algebra
Intermediate algebra often involves combining your knowledge of basic algebraic operations with more complex concepts like binomial multiplication and the difference of squares. These skills are foundational for higher-level math and critical for solving algebraic expressions. By recognizing patterns, such as the difference of squares, you simplify problems quickly and correctly. Practicing these techniques advances your algebra skills and helps tackle more intricate problems involving polynomials, factoring, and quadratic equations in algebra.

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