Solution - Binomial expansion

inital addition

27 x^{3} + 135 x^{2} + 225 x + 125

27x^3+135x^2+225x+125

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Step-by-step explanation

1. Identify the coefficients

Use the standard form of a binomial theorem ${(1 + x)}^{n} = \sum_{k = 0}^{\infty} (\frac{n}{k}) x^{k}$ ,to find the coefficients:
${(3 x + 5)}^{3}$
$a = 3 x$ , $b = 5$ , $n = 3$

2. Expand using binomial theorem

Let's expand each term step-by-step.

Let's plug in the values into the binomial expansion formula

Let’s simplify each term of the expansion

Finally, add all the terms together to get the expanded form

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Why learn this

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Why would you want to learn binomial expansion? Well, if you're planning the next big marketing campaign and want to predict how your customer base will grow, binomial expansion is exactly what you need! Let’s say you estimate that each month, 10% of your current customers will bring in new ones. To figure out how much your business will grow over 3 months, you can expand (1 + 0.1)^3:

${(1 + 0.1)}^{3} = 1 + 3 (0.1) + 3 {(0.1)}^{2} + {(0.1)}^{3} = 1 + 0.3 + 0.03 + 0.001 = 1.331$

So, after 3 months, your customer base is up by 33.1%. Look at you, expanding your business with math! Binomial expansion isn’t just for numbers—it’s a real-world tool to predict and plan for growth, making you a marketing pro in no time. Pretty cool, right?

Terms and topics

Binomial expansion