Algebra Expansion of (a + b)³

Questions and Answers

What is the expanded form of $(a + b)³$?

$a^3 + 2ab + b^3$

$a^3 + 3a^2b + 3ab^2 + b^3$ (correct)

$3a^2 + 3b^2$

$a^3 + b^3$

Which term in the expansion of $(a + b)³$ represents the product of $a$ and $b$?

$3ab$

$3a^2b$ (correct)

$2ab$

$3ab^2$ (correct)

What is the coefficient of the $a^2b$ term in the expansion of $(a + b)³$?

3 (correct)

If $a = 1$ and $b = 1$, what is the value of $(a + b)³$?

<p>8</p> Signup and view all the answers

Which of the following is a valid application of $(a + b)³$?

<p>Expanding the polynomial in two variables</p> Signup and view all the answers

Study Notes

Expansion of $(a + b)³$

The expanded form of $(a + b)³$ is given by the formula: [(a + b)³ = a³ + 3a²b + 3ab² + b³]
This expansion involves combining terms with varying powers of (a) and (b).

Term Representing the Product of $a$ and $b$

The term in the expansion that represents the product of (a) and (b) is (3a²b).
This term indicates that (a) is squared while (b) appears linearly, embodying their mixed product.

Coefficient of the $a²b$ Term

The coefficient of the (a²b) term in the expansion is (3).
This reflects how many ways we can select one (b) from the three total factors of ((a + b)) when expanding.

Value of $(a + b)³$ When (a = 1) and (b = 1)

If (a = 1) and (b = 1), then: [(1 + 1)³ = 2³ = 8]
This demonstrates the calculation of the cubic expansion at specific input values.

Valid Application of $(a + b)³$

The expression $(a + b)³$ can be applied in various algebraic contexts, including polynomial expansions, binomial theorem applications, and combinatorial problems.
It proves useful in solving real-world problems involving volume calculations when combining different quantities.

Description

Test your knowledge on the expansion of the binomial expression (a + b)³. Answer questions about the coefficients, terms, and specific values when a and b are given. Explore key applications of the binomial theorem as you delve into this algebra quiz.