Problem 36 - A Graphical Approach to College Algebra

Question

Find each product. $$(r-3 s+t)(2 r-s+t)$$

Step-by-Step Solution

Answer

The product is \(2r^2 + 3rt - 7rs + 3s^2 - 4st + t^2\).

Step 1: Apply the Distributive Property

Use the distributive property, also known as the FOIL method for binomials: \((r-3s+t)(2r-s+t) = (r)(2r-s+t) - (3s)(2r-s+t) + (t)(2r-s+t)\). Now, distribute each term in the first binomial across the second binomial.

Step 2: Distribute the First Term \(r\)

Distribute \(r\) across each term in the second binomial: \[(r)(2r-s+t) = r\cdot 2r - r\cdot s + r\cdot t = 2r^2 - rs + rt\].

Step 3: Distribute the Second Term \(-3s\)

Distribute \(-3s\) across each term in the second binomial: \[(-3s)(2r-s+t) = -3s\cdot 2r + 3s\cdot s - 3s\cdot t = -6sr + 3s^2 - 3st\].

Step 4: Distribute the Third Term \(t\)

Distribute \(t\) across each term in the second binomial: \[(t)(2r-s+t) = t\cdot 2r - t\cdot s + t\cdot t = 2rt - st + t^2\].

Step 5: Combine Like Terms

Combine all like terms from the previous steps: The expression is:\[2r^2 - rs + rt - 6sr + 3s^2 - 3st + 2rt - st + t^2\].Combine the like terms:\[2r^2 + (rt + 2rt) - rs - (6sr) + 3s^2 - (3st + st) + t^2\].Simplify:\[2r^2 + 3rt - 7rs + 3s^2 - 4st + t^2\].

Key Concepts

Distributive Property

The Distributive Property is a fundamental rule in algebra. It helps distribute one term across terms inside a parenthesis. This method allows us to simplify expressions and solve equations efficiently. You might think of it as spreading peanut butter over a slice of bread.
This property states:

• If you have an expression like \(a(b + c)\), you distribute \(a\) to \(b\) and \(c\) separately. The new expression is \(ab + ac\).
• In the given problem, we distribute each term in the first binomial \(r-3s+t\) over each term in the second binomial \(2r-s+t\).

By applying this property, you can simplify complex expressions and eventually solve them. Once you master the distributive property, you will find it easier to handle polynomial multiplication.

FOIL Method

The FOIL Method is specifically used for multiplying two binomials. FOIL is an acronym that stands for First, Outer, Inner, Last. These words represent the order in which you multiply the terms. Here's how it works:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms of the binomials.
• Last: Multiply the last terms of each binomial.

For the exercise, if it were a traditional binomial multiplication scenario, you would apply the FOIL method. However, here we use the distributive property repeatedly due to the trinomial expansion.
Understanding the FOIL method's order will make it simpler to follow a structured multiplication path and avoid missing any terms, especially in pairwise multiplications.

Binomials

A binomial is a polynomial with exactly two terms. In algebra, binomials are crucial because they are building blocks for more complex expressions. Each binomial in our problem can be expressed as:

• A binomial has the form \(a + b\), where \(a\) and \(b\) are terms which could involve numbers, variables, or both.
• In the exercise, we notice \(r-3s+t\) and \(2r-s+t\) seem to be composed of three terms, but each interaction here is tackled by considering pairs of terms for distribution, assisting in applying the binomial concept broadly.

Recognizing binomials helps in choosing the right multiplication method and simplifying the solution process.

Combining Like Terms

Combining Like Terms simplifies algebraic expressions by adding or subtracting terms with the same variable and exponent. This is an important step to make expressions more manageable:

• For instance, terms like \(3x\) and \(5x\) are "like" because they both contain the variable \(x\) raised to the same power. You can combine them as \(8x\).
• In the given exercise, after distributing, you end up with an expression that includes numerous like terms which need to be combined to simplify the expression.
• Look for all the terms with the same variables and exponents, and then add or subtract based on their coefficients.

In our example, combining like terms results in a neat expression: \[2r^2 + 3rt - 7rs + 3s^2 - 4st + t^2\].
Once you've mastered this skill, expressions with multiple terms will become simpler, paving your way for solving equations easily.