Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\((t-1)^{2}-49\)

Step-by-Step Solution

Answer

The expression \((t-1)^2 - 49\) factors to \((t - 8)(t + 6)\).

Step 1: Identify a and b

In the formula \((a^2 - b^2) = (a - b)(a + b)\), we can see that the expression \((t-1)^2 - 49\) fits this format with \(a = t-1\) and \(b = 7\).

Step 2: Apply the formula

To factor the expression, apply the formula for the difference of squares \((a^2 - b^2) = (a - b)(a + b)\). Replace \(a\) with \(t-1\) and \(b\) with \(7\) to get the factored form.

Step 3: Simplify

Simplify to find that \((t-1 - 7)(t-1 + 7) = (t-8)(t+6)\).

Key Concepts

Difference of squares

The difference of squares is a special pattern in algebra where you have two perfect squares separated by a subtraction sign. This is a useful concept because it allows us to factor expressions quickly and efficiently.

In general, this pattern is expressed as \(a^2 - b^2\), and it factors into the product \((a-b)(a+b)\). Recognizing this form in algebraic expressions helps simplify and solve equations more easily.

Take, for example, the expression \((t-1)^2 - 49\). Here, \((t-1)^2\) is the square of \(t-1\), and 49 is the square of 7. Hence, this expression is in the difference of squares form.

When you spot a difference of squares pattern like this, you can apply the formula directly to factor it. This makes it much easier than expanding or applying other lengthy methods.

Factoring polynomials

Factoring polynomials involves breaking down a complex polynomial into simpler expressions that, when multiplied together, provide the original polynomial. This is a fundamental skill in algebra and simplifies the process of solving equations.

There are several strategies for factoring polynomials, including:

• Finding the greatest common factor (if there is one).
• Applying special patterns such as the difference of squares.
• Identifying and using sum/difference of cubes or quadratic trinomials.
• Using factorization by grouping when applicable.

In our original example, \((t-1)^2 - 49\), we used the difference of squares to factor. Each factoring method fits specific types of problems and recognizing these types helps in choosing the correct technique.

Algebraic expressions

Algebraic expressions are combinations of variables, numbers, operations, and sometimes exponents that represent values and relationships. Understanding how to manipulate these expressions is crucial in solving algebra problems.

Working with algebraic expressions involves:

• Simplifying expressions using the order of operations.
• Combining like terms to make them easier to work with.
• Expanding expressions when necessary.
• Factorizing expressions to find their simplest forms.

In the expression \((t-1)^2 - 49\), for example, we have an algebraic expression that was initially set up in a special form—a difference of squares—allowing us to factor it efficiently into \((t-8)(t+6)\). Getting comfortable with identifying and manipulating these forms is key to mastering algebra.