Problem 6 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$9 a^{3} b(2 a-3 b+7 a b)$$

Step-by-Step Solution

Answer

The expanded product is: \(18a^4b - 27a^3b^2 + 63a^4b^2\).

Step 1: Understand the expression to be expanded

The given expression is a single monomial, \(9a^{3}b\), being multiplied by a trinomial, \((2a - 3b + 7ab)\). We need to distribute the monomial across the trinomial.

Step 2: Distribute the first term

Multiply \(9a^{3}b\) by the first term of the trinomial, \(2a\): \[ 9a^3b imes 2a = 18a^4b \].

Step 3: Distribute the second term

Multiply \(9a^{3}b\) by the second term of the trinomial, \(-3b\): \[ 9a^3b imes (-3b) = -27a^3b^2 \].

Step 4: Distribute the third term

Multiply \(9a^{3}b\) by the third term of the trinomial, \(7ab\): \[ 9a^3b imes 7ab = 63a^4b^2 \].

Step 5: Combine the distributed terms

Add the results from Steps 2, 3, and 4 to get the final expanded polynomial:\[ 18a^4b - 27a^3b^2 + 63a^4b^2 \].

Step 6: Simplify the expression

Combine like terms in the polynomial:Notice \(18a^4b\) and \(63a^4b^2\) have different terms, so they remain separate. The expression simplifies to:\[ 18a^4b + (63a^4b^2 - 27a^3b^2) \].

Key Concepts

Expanding Polynomials

Expanding polynomials is a fundamental skill in algebra that involves transforming a product of expressions into a sum or difference. This process is essential for simplifying expressions and solving equations. It generally requires you to apply the multiplication principle repeatedly. When expanding polynomials, you deal with different types of expressions, like monomials, binomials, or trinomials. In our particular example, a monomial, which is a single-term polynomial, is multiplied by a trinomial, a three-term polynomial. To expand, you follow these basic steps:

• Identify the expressions to multiply.
• Use the distributive property to multiply each term.
• Arrange and simplify the expression by combining like terms.

These steps convert the multiplication form into a streamlined polynomial, ready for further use or simplification.

Distributive Property

The distributive property is a vital algebraic rule used extensively when expanding polynomials. It states that for any real numbers, terms, or expressions, \(a(b + c) = ab + ac\). Essentially, you "distribute" the multiplier over each term in the parentheses.In the provided example, the term \(9a^3b\) is distributed across three separate terms in the trinomial \(2a - 3b + 7ab\). Here's a more detailed look at the steps:

• Multiply \(9a^3b\) by the first trinomial term \(2a\), yielding \(18a^4b\).
• Then, multiply \(9a^3b\) by the second term \(-3b\) to get \(-27a^3b^2\).
• Finally, multiply \(9a^3b\) by the third term \(7ab\), resulting in \(63a^4b^2\).

After distributing all terms, the initial expression is converted into an expanded form, which can be simplified by combining like terms or further manipulation, using algebraic principles.

Binomials and Trinomials

Understanding polynomials, especially binomials and trinomials, is crucial to mastering algebraic multiplication. Binomials are expressions containing two terms, such as \(x + y\), while trinomials consist of three terms, like \(x^2 + y^2 + z^2\).Calculating products involving binomials and trinomials requires systematic application of multiplication rules and algebraic identities. In our example, the focus is on multiplying a monomial with a trinomial, but understanding special cases with binomials helps strengthen these skills, such as:

• The Square of a Binomial: \((a + b)^2 = a^2 + 2ab + b^2\).
• Difference of Squares: \((a^2 - b^2) = (a + b)(a - b)\).

These identities and patterns can simplify the process significantly. Recognizing these forms can speed calculations and reduce errors when working with more complex polynomials in algebra.