Problem 76 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (t+8)(t-2) $$

Step-by-Step Solution

Answer

Answer: The product of (t+8)(t-2) after simplification is t^2 + 6t - 16.

Step 1: Apply the distributive property/FOIL method

To multiply the binomials, we will use the FOIL method, which stands for First, Outer, Inner, Last. This means we will multiply the first terms, the outer terms, the inner terms, and the last terms together, and sum the products. $$ (t+8)(t-2) = (t \cdot t) + (t \cdot (-2)) + (8 \cdot t) + (8 \cdot (-2)) $$

Step 2: Compute the products

Next, we will calculate each of the four products: $$ (t \cdot t) = t^2 \\ (t \cdot (-2)) = -2t \\ (8 \cdot t) = 8t \\ (8 \cdot (-2)) = -16 $$ Now we substitute these results back into the expression: $$ (t+8)(t-2) = t^2 - 2t + 8t - 16 $$

Step 3: Combine like terms

Finally, we will combine the like terms, which are the terms with the same variable raised to the same power. In this case, the like terms are the two terms with the variable \(t\): \(-2t\) and \(8t\). We will add these two terms together: $$ -2t + 8t = 6t $$ Now, we substitute this result back into our expression: $$ (t+8)(t-2) = t^2 + 6t - 16 $$ So, the result of the multiplication and simplification of the given binomials is: $$ (t+8)(t-2) = t^2 + 6t - 16 $$

Key Concepts

Distributive Property

The distributive property is an essential tool in algebra that helps you expand expressions and simplify equations. It states that when you multiply a sum by another number, you can distribute the multiplication over each term of the sum.

For instance, if you have an expression like \[(t+8)(t-2),\] this is a product of two binomials. The distributive property allows you to multiply each term in the first binomial by each term in the second binomial.

• Multiply the first term in the first binomial by every term in the second binomial.
• Repeat this process for the next term in the first binomial.

With this property, the expression unfolds into four individual multiplications. These steps lay the groundwork for understanding more complex algebraic manipulations.

FOIL Method

The FOIL method is a handy mnemonic that simplifies the distributive property when multiplying binomials. FOIL stands for First, Outer, Inner, Last, describing the order in which you multiply the terms.

Here's a breakdown of the FOIL method used on \[(t+8)(t-2):\]

• **First:** Multiply the first terms from each binomial: \[t imes t = t^2.\]
• **Outer:** Multiply the outermost terms: \[t imes -2 = -2t.\]
• **Inner:** Multiply the innermost terms: \[8 imes t = 8t.\]
• **Last:** Multiply the last terms from each binomial: \[8 imes -2 = -16.\]

The FOIL method is best for multiplying simple binomials quickly, as it ensures you've covered all combinations and transitions smoothly into combining like terms.

Like Terms

Like terms are terms within an algebraic expression that have identical variable components. To simplify an expression, we combine these terms by adding or subtracting their coefficients.

In the expression \[t^2 - 2t + 8t - 16,\] the like terms are \[-2t\] and \[8t.\] Both these terms have the variable \[t\] to the same power.

• Add their coefficients: \[-2 + 8 = 6.\]
• Combine them to create \[6t.\]

Once like terms are combined, the expression becomes simpler:\[t^2 + 6t - 16.\]Understanding how to identify and combine like terms is crucial for simplifying expressions, solving equations, and clarifying the true form of polynomial expressions.