Binomial Expansion Questions and Answers
The binomial theorem is a powerful tool for expanding expressions of the form $(x+y)^n$, where $n$ is a non-negative integer. It states that:
$(x+y)^n = \sum_{k=0}^n {n \choose k} x^{n-k} y^k$
where ${n \choose k}$ is the binomial coefficient, calculated as:
${n \choose k} = \frac{n!}{k!(n-k)!}$
Let's explore some examples and practice problems to solidify your understanding of binomial expansion.
Example 1: Expanding a Binomial to the Power of 3
Question: Expand $(x + 2)^3$ using the binomial theorem.
Solution:
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Identify the values: In this case, $x = x$, $y = 2$, and $n = 3$.
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Apply the binomial theorem:
$(x+2)^3 = {3 \choose 0} x^3 2^0 + {3 \choose 1} x^2 2^1 + {3 \choose 2} x^1 2^2 + {3 \choose 3} x^0 2^3$
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Calculate the binomial coefficients:
${3 \choose 0} = 1$, ${3 \choose 1} = 3$, ${3 \choose 2} = 3$, ${3 \choose 3} = 1$
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Simplify:
$(x+2)^3 = 1x^3 + 3x^2 * 2 + 3x * 4 + 1 * 8$
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Final result:
$(x+2)^3 = x^3 + 6x^2 + 12x + 8$
Example 2: Finding a Specific Term in a Binomial Expansion
Question: Find the term containing $x^5$ in the expansion of $(2x - 3)^8$.
Solution:
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Identify the relevant variables: Here, $x = 2x$, $y = -3$, and $n = 8$.
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Determine the power of $y$: Since we want the term with $x^5$, the power of $y$ must be $n - 5 = 3$.
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Apply the binomial theorem: The term containing $x^5$ is:
${8 \choose 3} (2x)^5 (-3)^3$
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Simplify:
$56 * 32x^5 * (-27) = -48384x^5$
Therefore, the term containing $x^5$ in the expansion of $(2x - 3)^8$ is $-48384x^5$.
Practice Problems
1. Expand $(x - 1)^4$ using the binomial theorem.
2. Find the term containing $y^6$ in the expansion of $(2 + y)^9$.
3. What is the coefficient of $x^3y^2$ in the expansion of $(x - 2y)^5$?
Answers:
1. $(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1$ 2. $84y^6$ 3. $80$
Remember, practice is key to mastering binomial expansion. Work through these examples and problems, and you'll be well on your way to understanding and applying the binomial theorem effectively.
