Question

explanation of all questions plzzzz​

Answer 1

1. It is true that,

Sum of coefficients in the expansion of (x + 1)ⁿ = 2ⁿ.

Because the coefficients alone can be got by taking x = 1. So (1 + 1)ⁿ = 2ⁿ.

Here it's given (x + 1)^8. Hence answer is 2^8 = 256.

2. First let me simplify it.

$\begin{aligned}&x^5+5x^4+10x^3+10x^2+5x+32\\\\\implies\ \ &x^5+5x^4+10x^3+10x^2+5x+1+31\\\\\implies\ \ &(x+1)^5+31\end{aligned}$

When x = -2,

$(-2+1)^5+31=(-1)^5+31=-1+31=\mathbf{30}$

3. Here also simplification is needed.

$\begin{aligned}&x^6-6x^5+15x^4-20x^3+15x^2-6x\\\\\implies\ \ &x^6-6x^5+15x^4-20x^3+15x^2-6x+1-1\\\\\implies\ \ &(x-1)^6-1\end{aligned}$

When $x=\sqrt[3]{2}+1,$

$(\sqrt[3]{2}+1-1)^6-1=(2^{\frac{1}{3}})^6-1=4-1=\mathbf{3}$

4. Here,

$\begin{aligned}&16x^4-32x^3+24x^2-8x-80\\\\\implies\ \ &16x^4-32x^3+24x^2-8x+1-81\\\\\implies\ \ &(2x-1)^4-3^4\\\\\implies\ \ &4((2x-1)^2+3^2)(x+1)(x-2)\end{aligned}$

Here we can see that x = 2 is a solution of $16x^4-32x^3+24x^2-8x-80=0.$

Hence, when x = 2, the polynomial will be 0.

5. No need to expand it. We've to find that in which polynomial there's only positive sign!

Because the given is (1 + 4x)^5, where both terms are positive.

Hence (2) is the answer.

6. In the expansion of (ax² + bx + c)ⁿ, the coefficient of x is $\mathrm{nbc^{n-1}}.$

Here,

a = 3 ; b = 2 ; c = 1 ; n = 10.

Hence answer is 10 · 2 · 1^9 = 20.