Problem 199 Multiply the following binomials... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 199

Multiply the following binomials. Use any method. $$(4 c-1)(4 c+1)$$

Short Answer

Expert verified

16c^2 - 1

Step by step solution

- Identify the binomials

Look at the given expression (4c - 1)(4c + 1). The two binomials are (4c - 1) and (4c + 1).

- Apply the Difference of Squares Formula

The given expression (4c - 1)(4c + 1) is in the form of (a - b)(a + b), where a is 4c and b is 1. The difference of squares formula is (a - b)(a + b) = a^2 - b^2.

- Substitute values into the formula

Here, a = 4c and b = 1. Substitute these values into the formula: (4c - 1)(4c + 1) = (4c)^2 - 1^2.

- Simplify the expression

Simplify the squared terms: (4c)^2 = 16c^2 and 1^2 = 1. So the expression becomes 16c^2 - 1.

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Key Concepts

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

Difference of Squares

The difference of squares is a useful algebraic pattern. It involves expressions of the form \begin{equation*}(a - b)(a + b)\text{.}ewline \text{Difference of squares formula: } (a - b)(a + b) = a^2 - b^2ewline When such an expression is multiplied, the result is always the first term squared minus the second term squared. ewline In our exercise, $$(4c - 1)(4c + 1)$$ we identify a as 4c and b as 1. By using the formula, this yields $$(4c)^2 - 1^2\text{.}$$

Simplifying Expressions

Once you substitute the values into the difference of squares formula, the expression needs to be simplified. Unpacking the differences of squares: \begin{equation*}(4c)^2 - 1^2 = 16c^2 - 1.ewline This means we square the 4c part and the 1 part.Simplifying this gives us the final expression:$$(4c)^2 = 16c^2$$ $$(1)^2 = 1\text{, thus the expression becomes }= 16c^2 - 1\text{.}$$

Binomial Multiplication

Binomial multiplication involves multiplying two binomials. In general, binomials are algebraic expressions containing two terms.In our case,$(4c - 1)(4c + 1)$ represents the multiplication of the binomials $4c-1$ and $4c+1$.Matching the binomials to the structure (a - b)(a + b) helps us to utilize the difference of squares formula.This pattern of multiplication simplifies the process and saves time instead of expanding each term:*First: Multiply the first terms.*Outer: Multiply the outer terms together and then the inner terms.*Last: Multiply the last terms.e.g, $$(4c * 4c = 16c^2\text{ and} (-1*+1 = -1\text{.})$$Add the results together simplifies to $16c^2 - 1\text{.}$

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91影视

Multiply the following binomials. Use any method. $$(4 c-1)(4 c+1)$$

Short Answer

Step by step solution

- Identify the binomials

- Apply the Difference of Squares Formula

- Substitute values into the formula

- Simplify the expression

Key Concepts

Difference of Squares

Simplifying Expressions

Binomial Multiplication

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