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sdas [7]
2 years ago
15

Use the binomial theorem to compute (2x-1)^5

Mathematics
1 answer:
OlgaM077 [116]2 years ago
3 0

Answer:

The expended form of the provided expression is: 32x^5-80x^4+80x^3-40x^2+10x-1

Step-by-step explanation:

Consider the provided expression.

(2x-1)^5

The binomial theorem:

(a+b)^{n}=\sum _{r=0}^{n}{n \choose r}a^{n-r}b^r

Where,

{n \choose r}= ^nC_r =\frac{n!}{(n-r)!r!}

Now by using the above formula.

\frac{5!}{0!\left(5-0\right)!}\left(2x\right)^5\left(-1\right)^0+\frac{5!}{1!\left(5-1\right)!}\left(2x\right)^4\left(-1\right)^1+\frac{5!}{2!\left(5-2\right)!}\left(2x\right)^3\left(-1\right)^2+\frac{5!}{3!\left(5-3\right)!}\left(2x\right)^2\left(-1\right)^3+\frac{5!}{4!\left(5-4\right)!}\left(2x\right)^1\left(-1\right)^4+\frac{5!}{5!\left(5-5\right)!}\left(2x\right)^0\left(-1\right)^5

2^5\cdot \:1\cdot \:1\cdot \:x^5-1\cdot \frac{2^4\cdot \:5x^4}{1!}+1\cdot \frac{2^3\cdot \:20x^3}{2!}-1\cdot \frac{2^2\cdot \:20x^2}{2!}+1\cdot \frac{5\cdot \:2x}{1!}+1\cdot \frac{\left(-1\right)^5}{\left(5-5\right)!}

32x^5-80x^4+80x^3-40x^2+10x-1

Hence, the expended form of the provided expression is: 32x^5-80x^4+80x^3-40x^2+10x-1

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