Problem 29 - A Graphical Approach to College Algebra

Question

Find each product. $$(5 m-6)(3 m+4)$$

Step-by-Step Solution

Answer

The product is \(15m^2 + 2m - 24\).

Step 1: Use the Distributive Property

To find the product \((5m-6)(3m+4)\), use the distributive property to multiply each term of the first binomial by each term of the second binomial: \[(a-b)(c+d) = ac + ad - bc - bd\].

Step 2: Multiply the First Terms

Multiply the first term of each binomial: \(5m \cdot 3m = 15m^2\).

Step 3: Multiply the Outer Terms

Multiply the outer terms together: \(5m \cdot 4 = 20m\).

Step 4: Multiply the Inner Terms

Multiply the inner terms: \(-6 \cdot 3m = -18m\).

Step 5: Multiply the Last Terms

Multiply the last terms: \(-6 \cdot 4 = -24\).

Step 6: Combine Like Terms

Now, combine all these results into a single expression: \[15m^2 + 20m - 18m - 24\]. Combine like terms: \[15m^2 + (20m - 18m) - 24 = 15m^2 + 2m - 24\].

Step 7: Write the Final Expression

The simplified expression from the multiplication is \[15m^2 + 2m - 24\]. This is the final product of the given binomials.

Key Concepts

Multiplying Binomials

Multiplying binomials involves using the distributive property to expand the expression into a quadratic equation. When you encounter binomials like

• \((a + b)(c + d)\),
• or in our case, \((5m - 6)(3m + 4)\),

you apply the distributive property by multiplying each term in the first binomial with each term in the second. This process can also be remembered with the FOIL method, which stands for:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the first term of the first binomial with the second term of the second binomial.
• Inner: Multiply the second term of the first binomial with the first term of the second binomial.
• Last: Multiply the last terms in each binomial.

To solve:

• Multiply \(5m \cdot 3m\) which gives \(15m^2\).
• Multiply \(5m \cdot 4\) which results in \(20m\).
• Multiply \(-6 \cdot 3m\) which is \(-18m\).
• Multiply \(-6 \cdot 4\) which equals \(-24\).

This transforms the expression from

• \((5m - 6)(3m + 4)\)

into

• \[15m^2 + 20m - 18m - 24\].

Combining Like Terms

After successfully multiplying the binomials, the next crucial step is to combine like terms. This means combining terms that have the same variable raised to the same power. In our expression

• \[15m^2 + 20m - 18m - 24\],

we need to look for terms that can be grouped:

• \(20m\) and \(-18m\) are like terms because they both have \(m\) as the variable.

Combining these gives us:

• \(20m - 18m = 2m\).

Thus, our simplified expression becomes:

• \[15m^2 + 2m - 24\].

By combining like terms, you simplify the expression, eliminating unnecessary complexity and making it easier to work with in future steps. This technique is key in solving algebraic expressions efficiently.

Algebraic Expressions

An algebraic expression consists of variables, constants, and arithmetic operations. In our example,

• \[15m^2 + 2m - 24\],

we have:

• The variable \(m\), which is raised to different powers (\(m^2\) and \(m\)).
• The constant \(-24\), which is a standalone number without a variable.

This expression combines these components using addition and subtraction. When dealing with algebraic expressions, it’s important to follow a systematic approach to combine, expand, or factor them efficiently.
During multiplication or any operation involving algebraic expressions:

• Pay close attention to signs when multiplying variables and constants together.
• Use parentheses to indicate the order of operations clearly when necessary.

Successfully managing these expressions lies at the core of solving more complex algebraic problems, enabling us to explore deeper areas in mathematics like calculus and beyond.