Problem 17 - Precalculus: Functions and Graphs

Question

Express as a polynomial. $$(3 a-5 b)(2 a+7 b)$$

Step-by-Step Solution

Answer

The polynomial is \(6a^2 + 11ab - 35b^2\).

Step 1: Use the Distributive Property

Apply the distributive property to expand the given expression \((3a - 5b)(2a + 7b)\). This means we'll multiply each term in the first set of parentheses by each term in the second set of parentheses: \(3a \times 2a + 3a \times 7b - 5b \times 2a - 5b \times 7b\).

Step 2: Perform Multiplications

Now, perform the multiplications: - \(3a \times 2a = 6a^2\)- \(3a \times 7b = 21ab\)- \(-5b \times 2a = -10ab\)- \(-5b \times 7b = -35b^2\)Collect these results: \(6a^2 + 21ab - 10ab - 35b^2\).

Step 3: Combine Like Terms

Combine the like terms from the expanded expression. The terms \(21ab\) and \(-10ab\) are like terms. Combine them to get: \(6a^2 + (21ab - 10ab) - 35b^2 = 6a^2 + 11ab - 35b^2\).

Step 4: Write the Polynomial

The polynomial expression for the product \((3a - 5b)(2a + 7b)\) is: \[6a^2 + 11ab - 35b^2\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, crucial for simplifying and solving equations. It enables you to remove parenthesis by distributing multiplication over addition or subtraction within an expression. In the given expression \((3a - 5b)(2a + 7b)\), the distributive property allows us to multiply each term in the first parenthesis by each term in the second.

• Think of it like distributing a "multiplier" to each part of another group.
• So here: \(3a\) is multiplied by both \(2a\) and \(7b\), and similarly \(-5b\) is multiplied by each term in the second group.

It's very orderly. Each term pairs off so nothing is left out. After applying the property, you'll notice that our expression turns into four multiplication operations: \(3a \times 2a, 3a \times 7b, -5b \times 2a,\) and \(-5b \times 7b\). This systematic approach ensures every combination of terms is considered, laying the groundwork to solve or simplify further.

Like Terms

Like terms are terms within an expression that have the exact same variable parts raised to the same power. Identifying and combining like terms is essential for simplifying expressions. In polynomials, this often reduces the complexity and amount of terms.

• For example, in the expanded polynomial \(6a^2 + 21ab - 10ab - 35b^2\), the terms \(21ab\) and \(-10ab\) are like terms because they both contain the same variables \(a\) and \(b\) raised to the same power.
• Unlike terms, on the other hand, do not share all the same variable factors.

By focusing on like terms, we can combine them by adding or subtracting their coefficients. For our polynomial, merging \(21ab - 10ab\) results in \(11ab\). This step is key in simplifying any polynomial expression because it reduces redundancy and streamlines the expression to its simplest form.

Expanding Expressions

Expanding expressions is the process of transforming a factored form of an algebraic expression into a full polynomial by distributing, multiplying, and then combining like terms. This is often necessary to facilitate further simplification or equation solving.

• Initially, an expression like \((3a - 5b)(2a + 7b)\) is compact and contained within its parentheses.
• But to expand it, you apply the distributive property, as previously described, to expose and rearrange all the multiplication interactions.

From this example, you ended up with \(6a^2 + 21ab - 10ab - 35b^2\). This new form, though busier, practically unfolds the hidden detail in the original expression, showing exactly how each piece relates to the others.Ultimately, expanding expressions not only helps in solving equations but also deepens understanding of the individual components of expressions.