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Magazine Chapter 4: Problem 88
Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (8 m+3 n)^{2} $$
Short Answer
Expert verified
(8m + 3n)^2 = 64m^2 + 48mn + 9n^2
Step by step solution
01
- Identify the binomial
Recognize that \(8m + 3n\) is the binomial that needs to be squared.
02
- Apply the formula for squaring a binomial
Use the formula \[ (a + b)^2 = a^2 + 2ab + b^2 \] where \(a = 8m\) and \(b = 3n\).
03
- Calculate a^2
Compute \(a^2\): \[ (8m)^2 = 64m^2 \]
04
- Calculate 2ab
Compute \(2ab\): \[ 2 \times 8m \times 3n = 48mn \]
05
- Calculate b^2
Compute \(b^2\): \[ (3n)^2 = 9n^2 \]
06
- Combine the results
Combine the results from steps 3, 4, and 5: \[ (8m + 3n)^2 = 64m^2 + 48mn + 9n^2 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Squaring a Binomial
Squaring a binomial is a foundational concept in algebra that involves multiplying a binomial by itself. To break it down, suppose you have a binomial expression \(a + b\), which you want to square. Here is the step-by-step formula: \[ (a + b)^2 = a^2 + 2ab + b^2 \]
This formula shows that when you square a binomial, you'll add three specific terms:
\[ (8m + 3n)^2 = (8m)^2 + 2(8m)(3n) + (3n)^2 \]
When you compute these, you get:
This formula shows that when you square a binomial, you'll add three specific terms:
- \(a^2\): The first term squared.
- \(2ab\): Twice the product of the two terms.
- \(b^2\): The second term squared.
\[ (8m + 3n)^2 = (8m)^2 + 2(8m)(3n) + (3n)^2 \]
When you compute these, you get:
- \(64m^2\)
- \(48mn\)
- \(9n^2\)
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations (such as +, -, *, and /). A few key points to remember:
- Algebraic expressions can include constants (fixed numbers like 3 or -5), variables (letters that represent unknown values, like m and n), and coefficients (numbers multiplied by variables).
- Simple examples include terms like \(8m\) or \(3n\), where 8 and 3 are coefficients, m and n are variables.
- More complex algebraic expressions can combine multiple terms, like \(64m^2 + 48mn + 9n^2\).
- Understanding how to manipulate algebraic expressions is key for solving equations and performing operations like squaring a binomial.
Binomial Theorem
The Binomial Theorem provides a fast way to expand binomials raised to a power. For squaring a binomial, this special case is straightforward and uses the formula \[ (a + b)^2 = a^2 + 2ab + b^2 \].
Generally, the Binomial Theorem says:
\[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\]\, where \( \binom{n}{k}\) are binomial coefficients.
For instance, with \(n=2\) and \(a=8m\), \(b=3n\), we use it to expand: \[ (8m + 3n)^2 = \binom{2}{0}(8m)^2(3n)^0 + \binom{2}{1}(8m)^1(3n)^1 + \binom{2}{2}(8m)^0(3n)^2 \]\
Generally, the Binomial Theorem says:
\[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\]\, where \( \binom{n}{k}\) are binomial coefficients.
For instance, with \(n=2\) and \(a=8m\), \(b=3n\), we use it to expand: \[ (8m + 3n)^2 = \binom{2}{0}(8m)^2(3n)^0 + \binom{2}{1}(8m)^1(3n)^1 + \binom{2}{2}(8m)^0(3n)^2 \]\
- \( \binom{2}{0} = 1, (8m)^2 = 64m^2, (3n)^0 = 1 \)
- \( \binom{2}{1} = 2, (8m)^1= 8m, (3n)^1 = 3n \)
- \( \binom{2}{2} = 1, (8m)^0=1, (3n)^2 = 9n^2 \)
Polynomials
Polynomials are algebraic expressions formed by multiple terms, typically involving variables raised to different powers. Key aspects of polynomials include:
- Each term in a polynomial consists of a coefficient, a variable, and an exponent (e.g., \(64m^2\), where 64 is the coefficient, m is the variable, and 2 is the exponent).
- Polynomials can have constants (like \(9n^2\)) and may contain multiple terms, giving rise to various degrees of polynomials. The degree is the highest exponent present (e.g., 2, for \(64m^2 + 48mn + 9n^2\)).
- Common operations include addition, subtraction, multiplication, and division of polynomials.
