Question

$$2 a b\left(9 a^{2}+6 a b-3 b^{2}\right)$$

Short answer

\(18a^3b + 12a^2b^2 - 6ab^3\)

Step-by-step solution

Distribute the outer term to each term inside the parentheses

Multiply the term outside the parentheses, which is \(2ab\), with each term inside the parentheses. The expression inside the parentheses is \((9a^2 + 6ab - 3b^2)\). This involves three separate multiplications: \(2ab \times 9a^2\), \(2ab \times 6ab\), and \(2ab \times (-3b^2)\).

Apply the Distributive Property

Perform each multiplication: For the first term, \(2ab \times 9a^2\) becomes \(18a^3b\). For the second term, \(2ab \times 6ab\) becomes \(12a^2b^2\). For the third term, \(2ab \times (-3b^2)\) becomes \(-6ab^3\). Combine the results of these multiplications to obtain the expanded form.

Write the final expression

Combine the results of the multiplications to get the final expanded expression: \(18a^3b + 12a^2b^2 - 6ab^3\).

Key concepts

Algebraic Expression Multiplication

Multiplying algebraic expressions involves combining like terms and applying certain rules of arithmetic to variables just as you would with numbers. In algebra, when you're given an expression such as 2ab(9a^2 + 6ab - 3b^2), you're faced with the task of not just simple multiplication, but distribution across multiple terms.

Multiplication of algebraic expressions extends beyond the basic arithmetic of numbers. It incorporates the concept of variables represented by letters which stand in for unknown quantities. When multiplying expressions, each term in one expression must be multiplied by each term in the other expression. This process is often referred to as the FOIL method when dealing with binomials. However, in cases with more terms, it's simply the process of ensuring that every term is accounted for in the multiplication process. It's crucial here to understand the laws of exponents for multiplying variables correctly. The operation of multiplying like bases results in their exponents being added together. For instance, multiplying a of power 1 with a^2 results in a^(1+2) or a^3.

When faced with such problems, a systematic approach to multiplying each term methodically can save time and reduce errors. Precision in each step is key to achieving the correct solution.

Polynomial Expansion

The process of expanding a polynomial such as (2x + 3)(x^2 - x + 4) involves applying multiplication to every term of the first polynomial by every term of the second polynomial. Product of this multiplication results in terms that might be like terms, which are then combined to reach a simplified expression.

In algebra, expanding polynomials is a fundamental skill that requires an understanding of not just the distributive property but also the ability to organize and combine like terms. Like terms are terms that have the same variables raised to the same power. After distributing and multiplying, significant importance lies in combining these like terms to simplify the expression. The result of a polynomial expansion is often a polynomial of a higher degree. For instance, expanding the expression (x + 1)(x - 2) yields a quadratic polynomial x^2 - x - 2.

Why Expand Polynomials?

• It allows for easier integration and differentiation in calculus.
• It aids in solving equations by finding roots or factors.
• It is vital in simplifying complex expressions.

Understanding the structure of polynomials and mastering the process of expansion are crucial steps towards solving more complex algebraic problems.

Applying the Distributive Property

The distributive property, also known as the distributive law, is a cornerstone of algebra that allows us to multiply a single term across a sum or difference within parentheses. Essentially, the distributive property tells us that a(b + c) = ab + ac, no matter the values of a, b, or c. This rule holds for any number of terms within the parentheses and any type of algebraic terms, including those with variables and exponents.

When you come across a problem like 2ab(9a^2 + 6ab - 3b^2), you have to distribute 2ab across each term inside the parentheses. Applying this property systematically ensures that no term is left behind and each multiplication is performed accurately. You multiply 2ab individually by 9a^2, 6ab, and -3b^2 to create an expanded expression.

Significance of the Distributive Property

• It provides a methodical approach to handling complex multiplication.
• Aids in simplifying expressions before solving equations.
• Helps in understanding polynomial expansion and factorization.

The distributive property is fundamental not just in algebra, but in advanced mathematics where it's used to simplify and solve diverse problems. By practicing this property, students gain a better logical understanding of equations and how to manipulate them effectively.