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Chapter 11: Problem 87

Expand: \(\left.(x+2 y)^{5} . \text { (Section } 10.5, \text { Examples } 2 \text { and } 3\right)\)

Short Answer

Expert verified
The expanded form of \((x + 2y)^5\) is \(x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5\)

Step by step solution

01

Identify the coefficients

In this case, \(a = x\) and \(b = 2y\). The power to which the binomial is raised, \(n\), is 5.
02

Apply the Binomial Theorem

Applying the binomial theorem to \((x + 2y)^5\) gives: \((a+b)^n = \binom{5}{0}x^5 (2y)^0 + \binom{5}{1}x^{4} (2y) + \binom{5}{2}x^{3} (2y)^2 + \binom{5}{3}x^{2} (2y)^3 + \binom{5}{4}x (2y)^4 + \binom{5}{5}x^0 (2y)^5\)
03

Compute the Binomial Coefficients

Next we calculate the binomial coefficients, which are: \(\binom{5}{0} = 1\), \(\binom{5}{1} = 5\), \(\binom{5}{2} = 10\), \(\binom{5}{3} = 10\), \(\binom{5}{4} = 5\), \(\binom{5}{5} = 1\)
04

Substitute the Coefficients

We substitute these coefficients and simplify to get: \(1 * x^5 * 1 + 5 * x^4 * 2y + 10 * x^3 * (2y)^2 + 10 * x^2 * (2y)^3 + 5 * x * (2y)^4 + 1 * (2y)^5\)
05

Simplify the Expression

Finally, we simplify the expression to obtain: \(x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5\)

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Key Concepts

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

Binomial Expansion
Binomial expansion is a process which allows us to expand expressions that are raised to a power. When you encounter a binomial like \((x+2y)^5\), this means an expression with two terms where each term can be raised to a certain power, in this case, 5. This concept is rooted deeply in the Binomial Theorem, a powerful algebraic method.

The Binomial Theorem states that:
  • \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
  • "\(a\)" and "\(b\)" are the terms of the binomial
  • \(n\) is the exponent to which the binomial is raised
  • \(\binom{n}{k}\) refers to the binomial coefficient, which determines the weight of each term in the expansion
These coefficients can be calculated using combinations, which tell us how many ways we can select a set of terms. By applying the Binomial Theorem, we transform complex expressions into simpler terms.
Pascal's Triangle
Pascal's Triangle is a fascinating mathematical concept that provides a simple and visual way to find the coefficients in a binomial expansion. Picture it as a triangular array where each number is the sum of the two numbers directly above it.

For example, the first few rows of Pascal鈥檚 Triangle look like this:
  • Row 0: 1
  • Row 1: 1, 1
  • Row 2: 1, 2, 1
  • Row 3: 1, 3, 3, 1
  • Row 4: 1, 4, 6, 4, 1
  • Row 5: 1, 5, 10, 10, 5, 1
Pascal's Triangle is directly linked to the coefficients in the Binomial Theorem; the numbers in the \(n\)-th row align perfectly with the binomial coefficient \(\binom{n}{k}\). As a result, you can simply use the Triangle to quickly locate these coefficients. This makes finding the terms in a binomial expansion much more efficient and intuitive.
Polynomial Expansion
A polynomial expansion is the result of applying the binomial theorem to an expression like \((x+2y)^5\), splitting it into a polynomial format. The expansion process involves substituting each term in the expansion with its corresponding binomial coefficient and simplifying.

To visualize how polynomial expansion works, let's tackle \((x+2y)^5\) step-by-step:
  • After applying the binomial theorem, each term looks like \(\binom{5}{k}x^{5-k}(2y)^k\)
  • Simplification involves calculating powers of numbers (e.g., \((2y)^k\))
  • Each resulting term of the expansion is added together to form the full polynomial
As seen in our previous step-by-step, this gives us the simplified polynomial: \(x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5\). Recognizing this process helps in breaking down seemingly complicated expressions into a neat series of terms.

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Most popular questions from this chapter

The following piecewise function gives the tax owed, \(T(x)\) by a single taxpayer for a recent year on a taxable income of \(x\) dollars. $$T(x)=\left\\{\begin{array}{cl}0.10 x & \text { if } \quad 0 < x \leq 8500 \\\850.00+0.15(x-8500) & \text { if } \quad 8500 < x \leq 34,500 \\\4750.00+0.25(x-34,500) & \text { if } 34,500 < x \leq 83,600 \\\17,025.00+0.28(x-83,600) & \text { if } 83,600 < x \leq 174,400 \\\42,449.00+0.33(x-174,400) & \text { if } 174,400 < x \leq 379,150 \\\110,016.50+0.35(x-379,150) & \text { if } \quad \quad \quad x > 379,150\end{array}\right.$$ a. Determine whether \(T\) is continuous at 8500 . b. Determine whether \(T\) is continuous at \(34,500\). c. If \(T\) had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. \(f\) and \(g\) are both continuous at \(a\), although \(f+g\) is not.

Show that the \(x\) -coordinate of the vertex of the parabola whose equation is \(y=a x^{2}+b x+c\) occurs when the derivative of the function is zero.

Research and present a group report about the history of the feud between Newton and Leibniz over who invented calculus. What other interests did these men have in addition to mathematics? What practical problems led them to the invention of calculus? What were their personalities like? Whose version established the notation and rules of calculus that we use today?

Explain how to read \(\lim _{x \rightarrow a} f(x)=L\)
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