Question

Find the coefficient of x⁴ in the product (1+2x)⁴(2-x)5 using binomial theorem

Answer 1

Please see the attachment

Answer 2

Answer:

-438 !!

Step-by-step explanation:

We first expand each of the factors of the given product using binomial theorem. We have,

(1+2a)⁴ =

${}^{4} c_{0} {}^{4} c_{1}(2a) + {}^{4} c_{2} {(2a)}^{2} + {}^{4} c_{3} {(2a)}^{3} + {}^{4} c_{4} {(2a)}^{4}$

= 1 + 4(2a) + 6(4a²) + 4(8a³) + 16a⁴

= 1 + 8a + 24a² + 32 a³ + 16 a⁴.

and

${(2 - a)}^{5}$

${}^{5} c_{0} {(2)}^{5} - {}^{5} c_{1} {(2)}^{4}(a) + {}^{5} c_{2} </p><p>{(2)}^{3} {(a)}^{2} - {}^{5} c_{3} {(2)}^{2} {(a)}^{3} + {}^{5} c_{4}(2) {(a)}^{4} - {}^{5} c_{5} {(a)}^{5}$

= 32 - 80a + 80a² - 40a³ + 10a⁴ -a^5

Thus (1+2a)⁴(2-a)^5

=(1+8a+24a²+32a³+16a⁴)(32-80a+80a²-40a³+10a⁴-a^5)

The complete multiplication of the two brackets need not be carried out.

We write only those terms which involve a⁴. This can be done if we note that a^r.a^4-r = a⁴.

The terms containing a⁴ are

1(10a⁴)+(8a)(-40a³)+(24a²)(80a²)+(32a³)(-80a)+(16a⁴)+32)= -438a⁴.

Thus, the coefficient of a⁴ in the given product is -438.