Exercise · ZW-BINOMIAL-THEOREM-003 · ★★☆ Intermediate
Find the coefficient of \(x^3\) in \((x+2)^5\).
Choose one option.
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- D. 40
Explanation
The term is \(\binom{5}{3}x^3\cdot 2^2=10\cdot 4\,x^3=40x^3\). Forgetting the power of 2 gives 10; using \(2^3\) gives 80.
Review the concept: Binomial theorem
The binomial theorem expresses the power \((x+y)^n\), with \(n\) a nonnegative integer, as a sum of terms whose coefficients are the binomial coefficients \(\binom{n}{k}\).
\[(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}\]
More practice problems: binomial theorem
- ZW-BINOMIAL-THEOREM-001 Find the coefficient of x^2 in the expansion of (x+1)^5.
- ZW-BINOMIAL-THEOREM-002 How many terms does the expansion of (a+b)^8 have?
- ZW-BINOMIAL-THEOREM-004 Find the coefficient of x^2 in (2x-1)^4.
- ZW-BINOMIAL-THEOREM-005 What is ∑_k=0^7binom7k?
- ZW-BINOMIAL-THEOREM-006 In the expansion of (x+y)^6, what is the coefficient of the middle term x^3y^3?
- ZW-BINOMIAL-THEOREM-007 Expand (x-y)^3.
- ZW-BINOMIAL-THEOREM-008 Find the constant term of the expansion of (x+1x)^6.
- ZW-BINOMIAL-THEOREM-009 Compute ∑_k=0^8(-1)^kbinom8k.