How do I find missing values in binomial expansions?

Question

I have the equation $(1+2x)^a=b+6x+12x^c+dx^3$ , an I am asked to find the values of $a$ , $b$ , $c$ , and $d$ . How do I find these values?

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Answer 1 · 2018-06-25T20:08:28

See a solution process below:

$(1 + 2x)^color(red)(a) = color(blue)(b) + 6x +12x^color(green)(c) + color(purple)(d)x^3$

Because the large exponent on the right side of the equation is $3$ then $color(red)(a)$ will equal $color(red)(3)$ :

$(1 + 2x)^color(red)(3) = color(blue)(b) + 6x +12x^color(green)(c) + color(purple)(d)x^3$

The constant will be $1^3 = 1$ therefore: $color(blue)(b)$ will be $color(blue)(1)$

$(1 + 2x)^color(red)(3) = color(blue)(1) + 6x +12x^color(green)(c) + color(purple)(d)x^3$

Because this is a binomial the exponent $color(green)(c)$ will be $color(green)(2)$

$(1 + 2x)^color(red)(3) = color(blue)(1) + 6x +12x^color(green)(2) + color(purple)(d)x^3$

The coefficient of the $x$ term on the left side of the equation will be cube: $2^3 = 8$ therefore $color(purple)(d)$ will be $color(purple)(8)$

$(1 + 2x)^color(red)(3) = color(blue)(1) + 6x +12x^color(green)(2) + color(purple)(8)x^3$