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Chapter 2: Problem 16

Expand and combine like terms. $$ (x-y)^{3} $$

Short Answer

Expert verified
Answer: The expanded form of \((x-y)^3\) is \(x^3 - 3x^2y + 3xy^2 - y^3\).

Step by step solution

01

Apply the binomial theorem

To expand \((x-y)^3\), we will use the binomial theorem which states that: $$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$ In our case, n=3 and y is replaced by -y, so the expansion will be: $$(x-y)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} (-y)^k$$
02

Calculate the binomial coefficients and expand the terms

Now, we will calculate each term in the expansion by substituting the values of k and substituting the binomial coefficients: $$(x-y)^3 = \binom{3}{0} x^{3-0} (-y)^0 + \binom{3}{1} x^{3-1} (-y)^1 + \binom{3}{2} x^{3-2} (-y)^2 + \binom{3}{3} x^{3-3} (-y)^3$$ Here, the four binomial coefficients are: \(\binom{3}{0} = 1, \binom{3}{1} = 3, \binom{3}{2} = 3\) and \(\binom{3}{3} = 1\)
03

Simplify each term

Substituting the values of the binomial coefficients and simplifying each term, we get: $$(x-y)^3 = 1\cdot x^3\cdot (-y)^0 + 3\cdot x^2\cdot (-y)^1 + 3\cdot x^1\cdot (-y)^2 + 1\cdot x^0\cdot (-y)^3$$ $$(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$$ So the expanded form of \((x-y)^3\) is \(x^3 - 3x^2y + 3xy^2 - y^3\).

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Key Concepts

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

Polynomial Expansion
Polynomial expansion involves expressing a power of a binomial like \((x-y)^3\) as a sum of terms. Each term in this expansion is derived from applying the binomial theorem. The binomial theorem is a powerful tool that helps break down expressions into a simplified form.

The main goal is to rewrite a binomial raised to an integer power into a series of individual terms. This is useful because it allows us to see all the components of the binomial product more clearly. We start with a binomial \((a+b)^n\) and use the theorem formula:
  • \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)
This formula indicates that each term in the expansion is multiplied by a specific binomial coefficient, which we'll discuss more in another section.
Like Terms
When working with polynomial expansions, finding and combining like terms is crucial. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(x^3 - 3x^2y + 3xy^2 - y^3\), each term has unique variables and powers, making none of them like terms.

Combining like terms involves adding or subtracting coefficients of those terms. In our example, because no terms share the same variables and powers, there鈥檚 no need to combine鈥攖hey鈥檙e already simplified. However, in more complex polynomials, careful identification is needed to ensure simplification.
Binomial Coefficients
Binomial coefficients play a key role in determining the weights of each term in a polynomial expansion. They are the numbers found in Pascal's Triangle and are represented as \(\binom{n}{k}\). These coefficients tell us how many ways we can choose \(k\) elements out of \(n\), often referred to as "n choose k."

Let's look at the expansion of \((x-y)^3\). The binomial coefficients here are:
  • \(\binom{3}{0} = 1\)
  • \(\binom{3}{1} = 3\)
  • \(\binom{3}{2} = 3\)
  • \(\binom{3}{3} = 1\)
These coefficients scale the terms when expanding the polynomial. Each coefficient affects the corresponding term in the expansion. Understanding these helps in predicting and verifying the expanded form of binomials.

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Most popular questions from this chapter

Is the fraction \(\frac{x}{x+3}\) equivalent to \(1+\frac{x}{3} ?\)

Factor each expression completely. $$ (r+1)^{2}+12 t(r+1)+36 t^{2} $$

Rewrite the expression by taking out the common factors. \(6 r(s-2)-12(s-2)\)

Factor each expression completely. $$ (t+1)^{3}-25(t+1) $$

Write each expression as a sum, difference, or product of two or more algebraic fractions. There is more than one correct answer. Assume all variables are positive. $$ \frac{1}{x y z} $$
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