Problem 72 - Algebra A Combined Function

Question

Multiply. $$ (3 x+2)(x+4) $$

Step-by-Step Solution

Answer

The product is \(3x^2 + 14x + 8\).

Step 1: Distribute the First Term

To begin, distribute the first term of the first binomial, which is \(3x\), to each term in the second binomial \((x+4)\). This means you need to multiply \(3x\) by \(x\) and \(3x\) by \(4\). The results are: \((3x)\times(x) = 3x^2\) and \((3x)\times(4) = 12x\).

Step 2: Distribute the Second Term

Next, distribute the second term of the first binomial, \(2\), to each term in the second binomial \((x+4)\). Multiply \(2\) by \(x\) and \(2\) by \(4\). The results are: \((2)\times(x) = 2x\) and \((2)\times(4) = 8\).

Step 3: Combine Like Terms

Combine the results from the previous steps. The expression is now: \(3x^2 + 12x + 2x + 8\). Combine the like terms \(12x\) and \(2x\), which adds up to \(14x\). The final expression is \(3x^2 + 14x + 8\).

Key Concepts

Distributive Property

The distributive property is a useful algebraic tool for multiplication across addition. It's essentially applying multiplication to each term in a sum separately. For example, when you have an expression like

• \((a + b) \times c\), you carry out:
  • Multiply \(a\) by \(c\)
  • Multiply \(b\) by \(c\)
  • Then add the results together to get \(ac + bc\).

In our specific exercise, you have

• \((3x + 2)(x + 4)\). To solve this, use the distributive property in two parts:
  • Distribute \(3x\) to \(x\) and \(4\), resulting in \(3x^2 + 12x\).
  • Next, distribute \(2\) to \(x\) and \(4\), giving \(2x + 8\).

These steps show how the distributive property works to simplify the multiplication of binomials.

Binomials

A binomial is a polynomial with exactly two distinct terms. It's similar to a simple sentence in language but in algebraic form. Binomials can look like

• \((a + b)\)
• \((x - c)\)

The key is that it only has two parts. In our example,

• The binomials are \((3x + 2)\) and \((x + 4)\).

When multiplying two binomials, each term from the first binomial must be multiplied by every term from the second. This combination of terms is where the magic happens.
Understanding what a binomial is helps make sense of why multiplication is performed in steps. Each term in both binomials has to interact with every term from the other binomial, ensuring no part is left out.

Combining Like Terms

After performing multiplication using the distributive property, you'll often end up with multiple terms. Some of these terms are likely to be "like terms." So, what are like terms? These are terms that have the same variable raised to the same power.For instance,

• \(12x\) and \(2x\) are like terms because they both have the \(x\) variable to the same power of one.

In our exercise, once you've multiplied everything out, the resulting expression is

• \(3x^2 + 12x + 2x + 8\).
• To simplify, combine the like terms \(12x\) and \(2x\) into \(14x\).
• That leads us to a cleaner expression: \(3x^2 + 14x + 8\).

Combining like terms simplifies the expression and makes it look tidy while helping to unclutter your final result.