Problem 11 - Elementary and Intermediate Algebra

Question

Complete each solution. $$ \begin{aligned} (2 x+5)(3 x-2) &=2 x(3 x)-\quad(2)+\quad(3 x)-(2) \\ &=6 x^{2}-\quad+10 \\ &=6 x^{2}+\quad-10 \end{aligned} $$

Step-by-Step Solution

Answer

The completed solution is \(6x^2 + 11x - 10\).

Step 1: Distribute the first term

Multiply the terms from the first binomial: \(2x\) and \(3x\), then \(2x\) and \(-2\). This will give you: \(2x \cdot 3x = 6x^2\) and \(2x \cdot (-2) = -4x\).

Step 2: Distribute the second term

Multiply the terms from the second binomial: \(5\) and \(3x\), then \(5\) and \(-2\). This results in \(5 \cdot 3x = 15x\) and \(5 \cdot (-2) = -10\).

Step 3: Combine like terms

Add the products you found to complete the solution: - Combine \(6x^2\), \(-4x\), and \(15x\) into \(6x^2 + 11x\). - Lastly, append \(-10\). This gives the final expression: \(6x^2 + 11x - 10\).

Key Concepts

Binomial Multiplication

When dealing with polynomials, especially those containing two terms, we often use a method called binomial multiplication. This is a key concept in algebra that helps us to simplify expressions like those in the format

• \((a + b)(c + d)\).

Binomial multiplication isn't just about throwing numbers around. Instead, it involves a systematic process where each term from one binomial gets multiplied by each term in the other binomial.
Take the exercise:

• \((2x + 5)(3x - 2)\).

To solve it, each component of the first binomial needs to multiply with every component of the second. This is often implemented in two main steps: first, multiplying each term by the terms in the other expression. This way, no part of either binomial is skipped.
Remember, each pair of terms multiplied together results in separate terms of the final expression. Ensuring all pairs are calculated is key to getting the full expansion correct.

Distributive Property

The distributive property is all about spreading things out evenly. It’s one of the most fundamental properties in algebra that helps in simplifying expressions and solving equations.
The core idea behind the distributive property is that multiplication distributes over addition or subtraction. Mathematically, this means that

• \(a(b + c) = ab + ac\).

In the context of our problem, this property is applied when multiplying terms from each binomial with each other. For example, multiplier \(2x\) from

• \((2x + 5)(3x - 2)\)

gets distributed across both \(3x\) and \(-2\), meaning

• \(2x(3x - 2)\) becomes \(2x \cdot 3x = 6x^2\) and \(2x \cdot (-2) = -4x\).

This process ensures that every possible product is included in the final expansion. Using the distributive property step-by-step like this helps us ensure all terms are accounted for and set us up for an accurate simplification.

Combining Like Terms

After multiplying and expanding a polynomial using the distributive property, you’ll often have several terms that look similar. That’s where the process of combining like terms comes in.
For instance, in our example, after expanding

• \((2x + 5)(3x - 2)\),

you end up with terms \(6x^2\), \(-4x\), \(15x\), and \(-10\). By 'like terms,' we mean terms that have the same variable raised to the same power, such as \(x\) terms or numbers by themselves (constants).
To combine like terms:

• Add or subtract coefficients of those terms.

For the \(-4x\) and \(15x\) terms, you combine them as follows: \(-4x + 15x = 11x\). This process simplifies the expression to \(6x^2 + 11x - 10\).

• Note: Keep an eye out for each power of the variable because mixing can lead to errors.

The goal here is to streamline the expression into its simplest form by collecting all like terms, which makes it easier to read, input, or further solve any algebra problems.