Problem 45 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (9-4 t)(9+4 t) $$

Step-by-Step Solution

Answer

The product of \( (9-4t)(9+4t) \) is \( 81-16t^2 \).

Step 1: Identify the terms

Recognize the form of the product is \( (a - b)(a + b) \), with 'a' being '9' and 'b' being '4t'.

Step 2: Apply the formula

Start applying the formula \( (a - b)(a + b) = a^2 - b^2 \).

Step 3: Substitute the values of a and b into the formula

After substituting, we obtain \(9^2 - (4t)^2 \).

Step 4: Calculate the squares

Calculate the square of 9 and square of \(4t\) to get \( 81 - 16t^2 \).

Key Concepts

Difference of Squares

When working with algebraic expressions, it's crucial to recognize specific patterns, such as the "difference of squares." This is a well-known algebraic identity where an expression is structured as \((a-b)(a+b)\), resulting in the difference of two square terms. For example, \((9-4t)(9+4t)\) is one such example. This identity can be applied to simplify complex expressions quickly and is given by:

• Formula: \((a-b)(a+b) = a^2 - b^2\)

Here, the difference of squares results in the square of the first term minus the square of the second. In our example of \((9-4t)(9+4t)\), we identify \(a\) as \(9\) and \(b\) as \(4t\), leading us to the evaluation of \(9^2 - (4t)^2\). Recognizing and using this pattern simplifies calculations in many algebraic problems.

Binomials

A binomial is an algebraic expression that contains exactly two terms. They can take different forms by involving addition, subtraction, or multiplication of variables and constants. For instance, both \((9 - 4t)\) and \((9 + 4t)\) are binomials. Understanding binomials is essential because they are the building blocks of more complicated polynomial expressions and are frequently used in algebraic expansion and factorization.When multiplying two binomials, each term in the first binomial is multiplied by each term in the second binomial. This often appears in calculations involving the distributive property or special product formulas, such as the difference of squares. By writing and reorganizing expressions with binomials, many algebraic challenges become manageable. Remember, practicing the manipulation of these basic structures can improve problem-solving skills significantly.

Multiplication of Expressions

Multiplication of expressions, especially involving binomials, is a fundamental skill in algebra. The process may involve special product formulas like the difference of squares, but generally, it requires expanding each term until all possible combinations are exhausted.To multiply binomials, use the distributive property, also known as expanding or the FOIL method (First, Outer, Inner, Last), which helps assure that no terms are missed:

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

For example, multiplying \((9-4t)\) and \((9+4t)\) involves recognizing the special product, but applying FOIL or direct expansion would yield the same result: \(81 - 16t^2\). Practice with different expressions improves fluency in simplifying and solving algebraic equations through strategic multiplication.