Question

Use the binomial theorem to expand the expression. $$\left(x^{2}+1\right)^{3}$$

   Use the binomial theorem to expand the expression.
$$\left(x^{2}+1\right)^{3}$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 10, Problem 25

Instant Answer

Solved by Expert Joshua Washington

05/27/2020

Step 1

The binomial theorem states that $(a+b)^n = \sum_{i=0}^{n} \binom{n}{i} a^{n-i} b^{i}$, where $\binom{n}{i}$ is the binomial coefficient. In our case, $a = x^{2}$, $b = 1$, and $n = 3$. Show more…

Show all steps

lock

Thanks for your feedback!

Use the binomial theorem to expand the expression. $$\left(x^{2}+1\right)^{3}$$

Play audio

Feedback

Powered by NumerAI

Joshua Washington and 60 other subject Precalculus educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Polynomial Expansion

Polynomial expansion refers to the process of rewriting an expression like (x^2 + 1)^3 in an equivalent form as a sum of multiple terms. This process uses the binomial theorem to distribute the exponent across the terms in the binomial, resulting in a series where each term is a product of a binomial coefficient and the terms raised to appropriate powers.

Binomial Theorem

The binomial theorem is a formula that allows the expansion of expressions raised to a power. In general, it states that (a + b)^n can be expressed as the sum of terms involving binomial coefficients multiplied by a raised to decreasing exponents and b raised to increasing exponents.

Binomial Coefficients

Binomial coefficients, commonly represented as C(n, k) or as 'n choose k', are the combinatorial factors that multiply each term in the expansion of a binomial expression. They determine the weight of each term and are calculated based on the number of ways to choose k items from a set of n items.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later