Question

The expression $x^5+10x^{4}a+40x^3{a}^2+80x^2{a}^3+80x{a}^4+32a^5$ is equal to

• A. $(x+3a{)}^5$
• B. $(x+2a{)}^5$
• C. $(x+a{)}^5$
• D. $(x+4a{)}^5$

Answer 1

The correct option is B $(x+2a{)}^5$

Given
$x^5+10x^{4}a+40x^3{a}^2+80x^2{a}^3+80x{a}^4+32a^5$

We know that,
$(x+y{)}^n={}^n{C}_0{x}^n+{}^n{C}_1{x}^{n-1}y+\cdots +{}^n{C}_n{y}^n$

Comparing both expression, we get
$n=5{}^n{C}_1{x}^{n-1}y=10x^{4}a\Rightarrow {}^5{C}_1{x}^{4}y=10x^{4}a\Rightarrow 5y=10a\Rightarrow y=2a$

Therefore,
$x^5+10x^{4}a+40x^3{a}^2+80x^2{a}^3+80x{a}^4+32a^5=(x+2a{)}^5$