WAEC SSCE - Further Mathematics - 2006 (Objective)

Question 1 Report

Find the coefficient of $x^{4}$ in the binomial expansion of $(1 - 2x)^{6}$.

320

240

-320

-240

Answer Details

We can use the binomial theorem to find the coefficient of $x^{4}$ in the expansion of $(1 - 2x)^{6}$. The binomial theorem states that the $n$th power of a binomial $(a + b)^{n}$ can be expressed as the sum of the terms of the form $a^{n-r}b^{r}$, where $r$ ranges from 0 to $n$. In this case, the binomial is $(1 - 2x)$, so we can expand it using the binomial theorem as follows: \[(1 - 2x)^{6} = \sum_{r=0}^{6} {6 \choose r} (1)^{6-r} (-2x)^{r}\] where ${6 \choose r}$ is the binomial coefficient. To find the coefficient of $x^{4}$, we need to find the term where $r$ is 2. So, we can substitute $r = 2$ into the formula above to get: \[{6 \choose 2} (1)^{4} (-2x)^{2} = 15 \times 4x^{2} = 60x^{2}\] However, we are looking for the coefficient of $x^{4}$, not $x^{2}$. Since the exponent of $x$ is 2 in the term we just calculated, we need to multiply it by another power of $x^{2}$ to get $x^{4}$. This means we need to find the term where $r$ is 4. Substituting $r = 4$ into the formula above, we get: \[{6 \choose 4} (1)^{2} (-2x)^{4} = 15 \times 16x^{4} = 240x^{4}\] Therefore, the coefficient of $x^{4}$ in the binomial expansion of $(1 - 2x)^{6}$ is 240. Hence, the answer is 240.






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