/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 \((c+d)(c+d)\)... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 16

\((c+d)(c+d)\)

Short Answer

Expert verified
(c+d)^2 = c^2 + 2cd + d^2

Step by step solution

01

Title - Application of the Binomial Theorem

First, recognize that \( (c+d)(c+d) \) is a binomial expression and can be expanded using the binomial theorem: \( (a+b)^2 = a^2 + 2ab + b^2 \). Set \( a = c \) and \( b = d \).
02

- Expand the Expression

Apply the binomial theorem: \( (c+d)^2 = c^2 + 2cd + d^2 \).

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a powerful tool for expanding expressions that are raised to a power. It states that \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k \), where \binom{n}{k}\ is a binomial coefficient calculated as \frac{n!}{k!(n-k)!} this exercise, we looked at the specific case where \(n = 2\). Recognizing that \( (c+d)^2\) is a binomial, the Binomial Theorem lets us quickly expand it to \(c^2 + 2cd + d^2\). This helps you understand the structure and components of the expanded polynomial.
Polynomial Expansion
Polynomial Expansion involves rewriting a product of sums as a sum of terms. This might sound complex, but it's just about organizing the terms properly. When you apply the Binomial Theorem, you effectively perform Polynomial Expansion. For \ (c+d)(c+d) \, you get \(c^2 + 2cd + d^2 \), which is easier to work with. Each term in the expanded form corresponds to a combination of terms from the original polynomial. Understanding these patterns is crucial for more complex algebraic manipulations.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In our exercise, \( (c+d)(c+d) \) is an algebraic expression. The goal is to make it simpler or more comprehensible by expanding it. Once expanded, \(c^2 + 2cd + d^2 \) makes it easy to see individual parts. Mastering the manipulation of algebraic expressions, including using the Binomial Theorem and Polynomial Expansion, is key to succeeding in algebra. You will encounter these techniques frequently in more advanced mathematics.

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