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Chapter 6: Problem 45

Determine whether each of the following is a difference of squares. $$ -1+64 t^{2} $$

Short Answer

Expert verified
Yes, \(-1 + 64t^2\) is a difference of squares.

Step by step solution

01

Understand the Difference of Squares Format

A difference of squares can be identified by the form: \[ a^2 - b^2 = (a-b)(a+b) \]. The expression must consist of two squared terms separated by a subtraction sign.
02

Identify Squared Terms

For \(-1 + 64t^2\), identify if each term is a perfect square. Notice that this term can be rewritten to \[ 64t^2 - 1 \] by reorganizing it.
03

Write Each Term as a Square

Recognize that \[ 64t^2 \] is \[ (8t)^2 \] and \[ 1 \] is \[ (1)^2 \]. Thus, the expression can be written as \[ (8t)^2 - 1^2 \].
04

Confirm the Difference of Squares

Since the rearranged expression \[ 64t^2 - 1 \] fits the form \[ a^2 - b^2 \] with \[ a = 8t \] and \[ b = 1 \], it is indeed a difference of squares.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Think of them as sentences in the language of algebra. For example, in the expression \(64t^2 - 1\), \(64t^2\) and \(1\) are terms, where \(64t^2\) includes a coefficient (\(64\)) and a variable \(t\) raised to an exponent (\(2\)).

To make sense of algebraic expressions:
  • Identify the terms and their components
  • Understand the operations connecting them
  • Simplify if necessary by combining like terms or factoring
In our exercise, recognizing that \(64t^2 - 1\) can be written as \(64t^2 + (-1)\) helps identify it as an algebraic expression that may fit the difference of squares pattern.
Factoring
Factoring is the process of breaking down an algebraic expression into simpler expressions (factors) that when multiplied together give the original expression. It's an essential skill in algebra that simplifies complicated problems.

Let's consider our example \(64t^2 - 1\):
  • First, rewrite the expression in a way that helps recognize patterns. In this case, it can be written as \[64t^2 - 1\]
  • Identify the squared terms. Notice \(64t^2\) can be written as \[ (8t)^2 \] and \(1\) as \[ (1)^2 \]
  • Recognize the difference of squares pattern: \[ a^2 - b^2 = (a - b)(a + b) \]
  • Apply the pattern to the expression: Since \[64t^2 - 1 = (8t)^2 - (1)^2\], it factors into \[ (8t - 1)(8t + 1) \]
By factoring, you transform the expression into a product of binomials, making it easier to work with.
Perfect Squares
Recognizing perfect squares is crucial for identifying difference of squares problems. A perfect square is a number or expression that can be written as the square of an integer or another expression.

For example:
  • \(4\) is a perfect square because \(4 = 2^2\)
  • \(25\) is a perfect square because \(25 = 5^2\)
  • \( (8t)^2 \) is a perfect square because it equals \( 64t^2 \)
In our specific exercise:
  • The term \(64t^2\) is a perfect square since it can be written as \[ (8t)^2 \]
  • The term \(1\) is a perfect square since it is \[ 1^2 \]

Recognizing these perfect squares allows us to apply the difference of squares formula, confirming that our expression \[64t^2 - 1\] is indeed a difference of squares.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ m^{2}-2 m n+n^{2}-25 $$

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{2 a}-y^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9-x^{2}-2 x y-y^{2} $$

Akio concludes that since \(x^{2}-9=(x-3)(x+3)\) it must follow that \(x^{2}+9=(x+3)(x-3) .\) What mistake(s) is he making?

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ -64+m^{2} $$
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