The coefficient of x9 in the expansion of (1 + x + x2 + x3)3(1 - x)6 is

Question

The coefficient of x9 in the expansion of (1 + x + x2 + x3)3(1 - x)6 is (A) -7 (B) 7 (C) 9 (D) -9

Tardigrade Admin · Accepted Answer

Given expansion can be written as, (1 + x + x2 + x3)3(1 - x)3 (1 - x)3 = (1 - x4)3(1 - x)3 = (1 - 3x4 + 3x8 - x12) (1 - 3x + 3x2 - x3) ∴ The coefficient of x9 is 3(-3) = - 9.

Q. The coefficient of $x^{9}$ in the expansion of
$(1 + x + x^{2} + x^{3})^{3} (1 - x)^{6}$ is

$- 7$

$7$

$9$

$- 9$

Given expansion can be written as,
$(1 + x + x^{2} + x^{3})^{3} (1 - x)^{3} (1 - x)^{3}$
$= (1 - x^{4})^{3} (1 - x)^{3}$
$= (1 - 3 x^{4} + 3 x^{8} - x^{12}) (1 - 3 x + 3 x^{2} - x^{3})$
$∴$ The coefficient of $x^{9}$ is $3 (- 3) = - 9$ .

Q. The coefficient of x9 in the expansion of (1+x+x2+x3)3(1−x)6 is

−7

7

9

−9

Given expansion can be written as, (1+x+x2+x3)3(1−x)3(1−x)3 =(1−x4)3(1−x)3 =(1−3x4+3x8−x12)(1−3x+3x2−x3) ∴ The coefficient of x9 is 3(−3)=−9.

Q. The coefficient of $x^{9}$ in the expansion of
$(1 + x + x^{2} + x^{3})^{3} (1 - x)^{6}$ is

$- 7$

$7$

$9$

$- 9$

Given expansion can be written as,
$(1 + x + x^{2} + x^{3})^{3} (1 - x)^{3} (1 - x)^{3}$
$= (1 - x^{4})^{3} (1 - x)^{3}$
$= (1 - 3 x^{4} + 3 x^{8} - x^{12}) (1 - 3 x + 3 x^{2} - x^{3})$
$∴$ The coefficient of $x^{9}$ is $3 (- 3) = - 9$ .