Solving Binomial Problems
\(g(x)=(1+kx)^{10}\) where the coeff of \(x^3\) is \(15\)
- The coefficient of \(x^{3}= \left( \frac{10}{3} \right)(1)^7(kx)^{3}=120k^3x^3\)
- \(120k^3x^3=15x^3\)
- \(k^3x^3=\frac{15}{120}x^3\)
- \(k^{3=\frac{1}{8}} \therefore k=\frac{1}{2}\)
Find the coefficient of \(x^3\) in the expansion of \((3+x)^5\)
\(^5C_{3}\times 3^{2}\times x^{3} = 90x^3\)
Find the coefficient of \(x^3\) in the expansion of \((1-x)^6\)
\(^6C_{3}\times 1^{3}\times x^{3}=20x^3\)
Find the coefficient of \(x^3\) in the expansion of \((1+x)^{10}\)
\(^{10}C_{3}\times 1^7\times x^{3}=120x^3\)
Find the coefficient of \(x^3\) in the expansion of \((1+x)^{20}\)
\(^{20}C_{3} \times 1^{17} \times x^{3}=1140x^3\)
Find the coefficient of \(x^3\) in the expansion of \(\left( 1-\frac{1}{2}x \right)^6\)
\(^6C_{3}\times 1^3\times \frac{1}{2}x^{3}=-\frac{5}{2}x^3\)
In the expansion of \((1+x)^{30}\), the coefficients of \(x^9\) and \(x^{10}\) are \(p\) and \(q\) respectively. Find the value of \(\frac{q}{p}\)