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Francisca Rodden

Answered question

2021-12-26

How do I find the binomial expansion of

{(2 x + 1)}^{3}

Answer & Explanation

eninsala06

Beginner2021-12-27Added 37 answers

We must use our knowledge of the binomial expansion:
Method 1:
We can use:

{(x + 1)}^{n} = 1 + n x + \frac{n (n - 1)}{2!} x^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{3} + \dots

Substituting

n = 3

and x for

2 x \Rightarrow

{(2 x + 1)}^{3} = 1 + (3 \cdot 2 x) + \frac{3 \cdot 2}{2!} \cdot {(2 x)}^{2} + \frac{3 \cdot 2 \cdot 1}{3!} \cdot {(2 x)}^{3}

= 1 + 6 x + 12 x^{2} + 8 x^{3}

ramirezhereva

Beginner2021-12-28Added 28 answers

Method 2:
We can use:

{(A + B)}^{n} = A^{n} + (\begin{matrix} n \\ 1 \end{matrix}) A^{n - 1} B^{1} + (\begin{matrix} n \\ 2 \end{matrix}) A^{n - 2} B^{2} + \dots

Letting

A = 2 x

and

B = 1

for this circumstance:

{(2 x + 1)}^{3} = {(2 x)}^{3} + (\begin{matrix} 3 \\ 1 \end{matrix}) {(2 x)}^{2} (1) + (\begin{matrix} 3 \\ 2 \end{matrix}) {(2 x)}^{1} {(1)}^{2} + (\begin{matrix} 3 \\ 3 \end{matrix}) {(2 x)}^{0} {(1)}^{3}

= 8 x^{3} + 12 x^{2} + 6 x + 1

nick1337

Expert2022-01-08Added 777 answers

Explanation:
we use Pascal's triangle for the coefficients
$(a + b)^{2} \to 1, 2, 1$
$(a + b)^{3} \to 1, 3, 3, 1$
So we need 1,3,3,1
the powers of the terms will sum to 3
and startingthe first term will be 3 and descend in 1 s thus
$(2 x + 1)^{3} = 1 (2 x)^{3} + 3 (2 x)^{2} (1) + 3 (2 x) (1)^{2} + 1 \cdot 1^{3}$
$(2 x + 1)^{3} = 8 x^{3} + 12 x^{2} + 6 x + 1$

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