If $(3 x - 1)^{7} = a_{7} x^{7} + a_{6} x^{6} + a_{5} x^{5} + . . . + a_{0}$ (A) then $a_{7} + a_{6} + . . . a_{0} = ?$

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Solution

$(3 x - 1)^{7} = a_{7} x^{7} + a_{6} x^{6} + a_{5} x^{5} + . . . + a_{0}$ (A)
For finding $a_{7} + a_{6} + a_{5} + . . . a_{0},$
Put x=1 in equation (A),
i.e., [ $3 (1) - 1]^{7} = a_{7} + a_{6} + a_{5} + . . . a_{0}$
$\Rightarrow a_{7} + a_{6} + a_{5} + . . . a_{0} = 2^{7}$ .

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Q. If

(3 x - 1) = a_{7} x^{7} + a_{6} x^{6} + . . . . + a_{0}

, then

a_{7} + a_{6} + . . . + a_{0}

The equation $a_{8} x^{8} + a_{7} x^{7} + a_{6} x^{6} + . . . + a_{0} = 0$ has all its roots positive and real $(w h e r e a_{8} = 1, a_{7} = - 4, a_{0} = {\frac{1}{2}}^{8})$ , then

Q. The equation

a_{8} X^{8} + a_{7} X^{7} + a_{6} X^{6} + \dots + a_{0} = 0

has all its roots positive and real

(w h e r e a_{8} = 1, a_{7} = - 4, a_{0} = \frac{1}{2^{8}})

, then which of the following is/are true?

Q. If

(1 + x)^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{10} x^{10}

, then value

(a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})^{2} + (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})^{2}

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If (3x−1)7=a7x7+a6x6+a5x5+...+a0 (A) then a7+a6+...a0=?

(3x−1)7=a7x7+a6x6+a5x5+...+a0 (A) For finding a7+a6+a5+...a0, Put x=1 in equation (A), i.e., [3(1)−1]7=a7+a6+a5+...a0 ⇒a7+a6+a5+...a0=27.

If $(3 x - 1)^{7} = a_{7} x^{7} + a_{6} x^{6} + a_{5} x^{5} + . . . + a_{0}$ (A) then $a_{7} + a_{6} + . . . a_{0} = ?$

$(3 x - 1)^{7} = a_{7} x^{7} + a_{6} x^{6} + a_{5} x^{5} + . . . + a_{0}$ (A)
For finding $a_{7} + a_{6} + a_{5} + . . . a_{0},$
Put x=1 in equation (A),
i.e., [ $3 (1) - 1]^{7} = a_{7} + a_{6} + a_{5} + . . . a_{0}$
$\Rightarrow a_{7} + a_{6} + a_{5} + . . . a_{0} = 2^{7}$ .