Problem 54 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(t-2)\left(t^{2}+7 t+2\right)$$

Step-by-Step Solution

Answer

The product is \(t^3 + 5t^2 - 12t - 4\).

Step 1: Review the Distributive Property

To solve the product \((t-2)(t^2 + 7t + 2)\), we can use the distributive property which states that \(a(b+c) = ab + ac\). This property will allow us to distribute each term from the binomial across each term in the trinomial.

Step 2: Distribute Each Term

1. Distribute the first term \(t\) from \(t-2\):- \(t(t^2) = t^3\)- \(t(7t) = 7t^2\)- \(t(2) = 2t\)2. Distribute the second term \(-2\) from \(t-2\):- \(-2(t^2) = -2t^2\)- \(-2(7t) = -14t\)- \(-2(2) = -4\)

Step 3: Combine Like Terms

Now, combine all the terms:- The \(t^3\) term is alone: \(t^3\)- Combine the \(t^2\) terms: \(7t^2 - 2t^2 = 5t^2\)- Combine the \(t\) terms: \(2t - 14t = -12t\)- The constant term is: \(-4\)The expression now is: \(t^3 + 5t^2 - 12t - 4\).

Key Concepts

The Distributive Property

The distributive property is a fundamental principle in mathematics used to simplify expressions. It's especially handy when you're dealing with multiplication involving polynomials. In this context, we use the distributive property to expand expressions like

• Multiply each term in one polynomial by every term in another polynomial
• Represent expressions in an expanded form

Given an expression \((t-2)(t^2 + 7t + 2)\), the distributive property helps us to distribute the terms as follows:

• The first term \(t\) in \((t-2)\) multiplies each term in \((t^2 + 7t + 2)\), resulting in three products: \(t^3, 7t^2,\) and \(2t\)
• The second term \(-2\) in \((t-2)\) also multiplies each term in \((t^2 + 7t + 2)\), yielding: \(-2t^2, -14t,\) and \(-4\)

This approach breaks down the multiplication process into manageable parts, making it easier to handle.

Combining Like Terms

Once you've expanded the polynomial expression using the distributive property, the next logical step is to combine like terms. Like terms are terms that have the same variable raised to the same power.In our example, after distributing the terms, the expression becomes: \(t^3 + 7t^2 + 2t - 2t^2 - 14t - 4\).Now, it's time to combine them:

• The \(t^3\) term stands alone and cannot be combined with any other terms.
• For the \(t^2\) terms: Combine \(7t^2\) and \(-2t^2\) to get \(5t^2\).
• For the \(t\) terms: Combine \(2t\) and \(-14t\) to get \(-12t\).
• Finally, the constant term is \(-4\) and remains unchanged.

By combining like terms, we simplify the expression to its final form: \(t^3 + 5t^2 - 12t - 4\). This step is crucial for reducing polynomial expressions to a clear and concise result.

Special Product Patterns

Special product patterns are shortcuts that can be used to quickly expand binomial expressions without having to apply distribution in the traditional way. These patterns are derived from basic algebraic identities. However, sometimes, especially with more complex polynomials like our case, traditional distribution and combining like terms is more straightforward.The most common special product patterns include:

• The square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\)
• The difference of squares: \((a + b)(a - b) = a^2 - b^2\)

While these patterns do not apply directly to our given problem \((t-2)(t^2 + 7t + 2)\), understanding them can be helpful in recognizing opportunities for simplification in other contexts. Learning when and how to leverage these shortcuts can save time and effort, especially when dealing with more cumbersome expressions.