Find the coefficients of a^4 in the product (1+2a)^ 4 (2-a)^ 5 using binomial theorem.

We have, (1+2a)^ 4 (2-a)^ 5 = [ ^4 C_ 0 (2a )^ 0 + ^4 C_ 1 (2a )^1+ ^4 C_ 2 (2a )^ 2 + ^4 C_ 3 (2a )^ 3 + ^4 C_ 4 (2a )^ 4 ] [ ^5 C_ 0 (2)^ 5 (-a)^ 0 + ^5 C_ 1 (2)^ 4 (-a)^ 1 + ^5 C_ 2 (2)^ 3 (-a)^ 2 + ^5 C_ 2 (2)^ 2 (-a)^ 3 + ^5 C_ 1 (2)^ 1 (-a)^ 5 = [1+8a+24a^ 2 +32a^ 3 +16a^ 4 ] [32-80a+80a^ 2 -40a^ 3 +10a^ 4 -a^ 5 ] Coefficient of a^ 4 =10-320+1920-2560+512=-438

Find the coefficients of a^4 in the product (1+2a)^ 4 (2-a)^ 5 us…

Download App

Question

Find the coefficients of $a^4$ in the product $(1+2\text{a})^{4}(2-\text{a})^{5}$ using binomial theorem.

✓

Answer

We have,
$(1+2\text{a})^{4}(2-\text{a})^{5}$
$=\Big[{^\text{4}}\text{C}_{\text{0}}\big(2\text{a}\big)^{0}+{^\text{4}}\text{C}_{\text{1}}\big(2\text{a}\big)^1+{^\text{4}}\text{C}_{\text{2}}\big(2\text{a}\big)^{2}+{^\text{4}}\text{C}_{\text{3}}\big(2\text{a}\big)^{3}+{^\text{4}}\text{C}_{\text{4}}\big(2\text{a}\big)^{4}\Big]\\ \times\Big[{^\text{5}}\text{C}_{\text{0}}(2)^{5}(-\text{a})^{0}+{^\text{5}}\text{C}_{\text{1}}(2)^{4}(-\text{a})^{1}+{^\text{5}}\text{C}_{\text{2}}(2)^{3}(-\text{a})^{2}+{^\text{5}}\text{C}_{\text{2}}(2)^{2}(-\text{a})^{3}+{^\text{5}}\text{C}_{\text{1}}(2)^{1}(-\text{a})^{5}$
$=\Big[1+8\text{a}+24\text{a}^{2}+32\text{a}^{3}+16\text{a}^{4}\Big]\times\Big[32-80\text{a}+80\text{a}^{2}-40\text{a}^{3}+10\text{a}^{4}-\text{a}^{5}\Big]$
Coefficient of $\text{a}^{4}=10-320+1920-2560+512=-438$