Mathematics, 29.11.2019 18:31 UncleVictor5188

Use the binomial theorem to expand (d-4b)^3

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Answer from: ray3699

12bd^2+48b^2d-64b^3[/tex]

Explanation:

we have general formula for binomial expansion which is

$(a+b)^n=a^n+na^{n-1} b +\frac{n(n-1)}{2} a^{n-2}b^2+----------+b^n$

Since, we have to evaluate

(d-4b)^3

here a=d and b= -4band n=3

Substituting values in the formula we will get

$d^3+3d^{3-1}(-4b)+ \frac{(3)(3-1)}{2} d^{3-2}(-4b)^2+(-4b)^3 \\\\d^3-12bd^2+48b^2d-64b^3$

Answer from: haileysolis5

The binomial expansion of the expression ${\left({d - 4b} \right)^3}$ is $\boxed{{{\mathbf{d}}^{\mathbf{3}}}{\mathbf{ - 12b}}{{\mathbf{d}}^{\mathbf{2}}}{\mathbf{ + 48}}{{\mathbf{b}}^{\mathbf{2}}}{\mathbf{d - 64}}{{\mathbf{b}}^{\mathbf{3}}}}$ .

Further explanation:

It is given that the expression is ${\left({d - 4b}\right)^3}$ .

Now, the expansion of the expression is simplified using binomial theorem.

The binomial is a polynomial with only two terms. The binomial theorem is used to simplify the algebraic expansion of any power of a binomial expression.

Now, consider and as the numbers and the binomial expression becomes x + y . If the power is assumed to be , then the binomial expression is ${\left({x + y}\right)^n}$ .

The binomial expression ${\left({x + y}\right)^n}$ is expanded as follows:

For any positive integer ,the expanded form is given below.

${\left( {x + y}\right)^n} = {x^n} + \left(\begin{aligned}n\hfill\\1\hfill\\\end{gathered}\right){x^{n - 1}}y + \left(\begin{aligned}n\hfill\\2\hfill\\\end{aligned}\right){x^{n - 2}}{y^2}+\cdots+\left( \begin{aligned}n\hfill\\r \hfill\\\end{gathered}\right){x^{n - r}}{y^r}+\cdots + \left(\begin{aligned}\,\,\,n\hfill\\n - 1\hfill\\\end{aligned}\right)x{y^{n - 1}} + {y^n}$

For non-negative integers and

with $r\leqslant n$ , the expression is $\left(\begin{aligned}n\hfill\\r\hfill\\\end{aligned}\right) = \dfrac{{n!}}{{r!\left({n - r} \right)!}}$ where, $n! = n\cdot \left({n - 1}\right)\cdot\left( {n - 2}\right)\cdots 3\cdot 2\cdot 1$ .

The expression ${\left({d - 4b}\right)^3}$ is expanded using the binomial theorem as follows:

$\begin{aligned}(d-4b)^{3}&=\left[\dbinom{3}{0}(d)^{3}(-4b)^{0}+\dbinom{3}{1}(d)^{2}(-4b)^{1}+\dbinom{3}{2}(d)^{1}(-4b)^{2}+\dbinom{3}{3}(d)^{0}(-4b)^{3}+\right]\\&=\left[\dfrac{3!}{3!\cdot 0!}d^{3}(1)+\dfrac{3!}{1!\cdot 2!}(-d^{2}4b)+\dfrac{3!}{2!\cdot 1!}d(16b^{2})+\dfrac{3!}{3!\cdot 0!}d(1)(-64b^{3})\right]\end{aligned}$

Simplify the above equation as follows:

$\begin{aligned}{\left({d - 4b}\right)^3}&=\left[{{d^3} + 3\left({ - {d^2}4b} \right) + 3d\left( {16{b^2}}\right)+\left({ - 64{b^3}}\right)} \right]\\&={d^3} - 12b{d^2} + 48{b^2}d - 64{b^3}\\\end{aligned}$

Thus, the binomial expansion of the expression ${\left( {d - 4b}\right)^3}$ is $\boxed{{{\mathbf{d}}^{\mathbf{3}}}{\mathbf{ - 12b}}{{\mathbf{d}}^{\mathbf{2}}}{\mathbf{ + 48}}{{\mathbf{b}}^{\mathbf{2}}}{\mathbf{d - 64}}{{\mathbf{b}}^{\mathbf{3}}}}$ .

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Answer Details:

Grade: Junior High School

Subject: Mathematics

Chapter: Binomial Theorem

Keywords: Binomial theorem, linear equation, system of linear equations in two variables, ${\left({d - 4b}\right)^3}$ , expression, theorem, expansion

Answer from: Quest

the distance from any point on the parabola to the focus is the exact same as the distance from that point to the directrix.

if you could provide answer choices that might allow me to answer the question easier if my above answer isn't an option! (:

Answer from: Quest

i can you with this question sweetie!

step-by-step explanation:

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