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Chapter 5: Problem 47

Squares of Binomials Multiply. $$(5 c-2 d)^{2}$$

Short Answer

Expert verified
(5c - 2d)^2 = 25c^2 - 20cd + 4d^2

Step by step solution

01

Understand the Formula

Recall the formula for the square of a binomial: \[ (a - b)^2 = a^2 - 2ab + b^2 \].In this case, let \( a = 5c \) and \( b = 2d \).
02

Apply the Formula

Substitute the values of a and b into the binomial square formula: \[ (5c - 2d)^2 = (5c)^2 - 2(5c)(2d) + (2d)^2 \].
03

Simplify Each Term

Calculate each term separately:1. \((5c)^2 = 25c^2\)2. \(2(5c)(2d) = 20cd\)3. \((2d)^2 = 4d^2\)
04

Combine the Terms

Substitute back in the simplified terms:\[ (5c - 2d)^2 = 25c^2 - 20cd + 4d^2 \].

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Key Concepts

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

Binomial Expansion
Binomial expansion involves expanding an expression that is raised to a power. In this case, we are looking at the square of a binomial.
For any binomial \((a + b)\) raised to the power of 2, the standard formula is: \[ (a + b)^2 = a^2 + 2ab + b^2 \]
When the binomial includes subtraction instead of addition, the formula is slightly different: \[ (a - b)^2 = a^2 - 2ab + b^2 \]
Binomial expansion helps break down more complex expressions into simpler terms, making them easier to understand and calculate.
Polynomial Multiplication
Polynomial multiplication is the process of multiplying polynomial expressions. When multiplying two binomials, like in our original problem, it often helps to use distribution or the FOIL method (First, Outer, Inner, Last).
For example, with \((a - b)^2\), you distribute each term: \[ (a - b)(a - b) = a(a - b) - b(a - b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2 \]
This is how we derive the formula we use in our step-by-step solution. Understanding polynomial multiplication is crucial for solving and simplifying higher-order algebraic equations.
Algebraic Simplification
Algebraic simplification involves combining like terms and reducing expressions to their simplest form.
In our example: \((5c - 2d)^2\)
Breaking it down, we apply: \[ (5c)^2 = 25c^2, -2(5c)(2d) = -20cd, (2d)^2 = 4d^2 \]
We then combine these into: \[ 25c^2 - 20cd + 4d^2 \]
Simplification makes it easier to understand and manipulate algebraic expressions, providing a clear and concise result.
Intermediate Algebra
Intermediate algebra builds on basic algebra concepts with more complex equations and functions. This includes polynomial equations, binomial theorems, and advanced factoring.
Understanding the square of a binomial is an essential part of intermediate algebra. This knowledge helps in solving quadratic equations, graphing polynomial functions, and progressing to higher-level math subjects.

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Most popular questions from this chapter

For \(P(x)\) and \(Q(x)\) as given, find the following. $$ \begin{aligned} &P(x)=13 x^{5}-22 x^{4}-36 x^{3}+40 x^{2}-16 x+75\\\ &Q(x)=42 x^{5}-37 x^{4}+50 x^{3}-28 x^{2}+34 x+100 \end{aligned} $$ $$ 3[P(x)]-Q(x) $$

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Perform the indicated operations. $$ \left(x^{2}-4 x+7\right)+\left(3 x^{2}-9\right)-\left(x^{2}-4 x+7\right) $$
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