Use The Formula $a^2-b^2=(a+b)(a-b$\] To Factor Completely.What Are The Factors Of $36x^8-49$?A. $(9x^4-1)(4x^2+49$\]B. $(36x^4-7)(x^2+7$\]C. $(6x^4-7)(6x^4+7$\]D. $(6x^8-7)(6x+7$\]

Feb 27, 2025 by ADMIN 182 views

**Factoring the Difference of Squares: A Step-by-Step Guide**

Introduction

In algebra, the difference of squares is a fundamental concept that allows us to factorize expressions of the form $a^2 - b^2$ . This concept is based on the formula $a^2 - b^2 = (a + b)(a - b)$ . In this article, we will explore how to factor completely using the difference of squares formula, and we will apply this concept to factorize the expression $36x^8 - 49$ .

Understanding the Difference of Squares Formula

The difference of squares formula is a fundamental concept in algebra that allows us to factorize expressions of the form $a^2 - b^2$ . This formula states that:

a^2 - b^2 = (a + b)(a - b)

This formula can be applied to any expression of the form $a^2 - b^2$ , where $a$ and $b$ are any real numbers or variables.

Applying the Difference of Squares Formula

To factor completely using the difference of squares formula, we need to identify the values of $a$ and $b$ in the given expression. In this case, we have the expression $36x^8 - 49$ . We can rewrite this expression as:

36x^8 - 49 = (6x^4)^2 - 7^2

Now, we can apply the difference of squares formula by substituting $a = 6x^4$ and $b = 7$ :

(6x^4)^2 - 7^2 = (6x^4 + 7)(6x^4 - 7)

Therefore, the factors of $36x^8 - 49$ are $(6x^4 + 7)(6x^4 - 7)$ .

Comparing the Factors

Now, let's compare the factors we obtained with the options provided:

A. $(9x^4 - 1)(4x^2 + 49)$ B. $(36x^4 - 7)(x^2 + 7)$ C. $(6x^4 - 7)(6x^4 + 7)$ D. $(6x^8 - 7)(6x + 7)$

We can see that option C matches our result: $(6x^4 - 7)(6x^4 + 7)$ .

Conclusion

In this article, we explored how to factor completely using the difference of squares formula. We applied this concept to factorize the expression $36x^8 - 49$ and obtained the factors $(6x^4 + 7)(6x^4 - 7)$ . We compared our result with the options provided and found that option C matches our result.

Tips and Tricks

When applying the difference of squares formula, make sure to identify the values of $a$ and $b$ in the given expression.
Use the formula $a^2 - b^2 = (a + b)(a - b)$ to factorize expressions of the form $a^2 - b^2$ .
Be careful when simplifying expressions and make sure to check your work.

Practice Problems

Factor completely using the difference of squares formula: $25x^2 - 16$
Factor completely using the difference of squares formula: $9x^4 - 16$
Factor completely using the difference of squares formula: $49x^2 - 36$

References

[Algebra textbook]
[Online algebra resources]

Glossary

Difference of squares: A fundamental concept in algebra that allows us to factorize expressions of the form $a^2 - b^2$ .
Factor completely: To factor an expression into its simplest form using the difference of squares formula.
Difference of squares formula: A formula that states $a^2 - b^2 = (a + b)(a - b)$ .

Q: What is the difference of squares formula?

A: The difference of squares formula is a fundamental concept in algebra that allows us to factorize expressions of the form $a^2 - b^2$ . The formula states that:

a^2 - b^2 = (a + b)(a - b)

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you need to identify the values of $a$ and $b$ in the given expression. Then, you can substitute these values into the formula and simplify the expression.

Q: What are some common mistakes to avoid when applying the difference of squares formula?

A: Some common mistakes to avoid when applying the difference of squares formula include:

Not identifying the values of $a$ and $b$ correctly
Not substituting the values of $a$ and $b$ into the formula correctly
Not simplifying the expression correctly

Q: Can I use the difference of squares formula to factorize expressions with variables?

A: Yes, you can use the difference of squares formula to factorize expressions with variables. For example, if you have the expression $x^2 - 4$ , you can factor it using the difference of squares formula as follows:

x^2 - 4 = (x + 2)(x - 2)

Q: Can I use the difference of squares formula to factorize expressions with negative numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with negative numbers. For example, if you have the expression $9 - 16x^2$ , you can factor it using the difference of squares formula as follows:

9 - 16x^2 = (3 + 4x)(3 - 4x)

Q: How do I know if an expression can be factored using the difference of squares formula?

A: To determine if an expression can be factored using the difference of squares formula, you need to check if the expression is in the form $a^2 - b^2$ . If it is, then you can use the formula to factor the expression.

Q: What are some examples of expressions that can be factored using the difference of squares formula?

A: Some examples of expressions that can be factored using the difference of squares formula include:

$x^2 - 4$
$9 - 16x^2$
$36x^8 - 49$
$25x^2 - 16$

Q: What are some examples of expressions that cannot be factored using the difference of squares formula?

A: Some examples of expressions that cannot be factored using the difference of squares formula include:

$x^2 + 4$
$9 + 16x^2$
$36x^8 + 49$
$25x^2 + 16$

Q: Can I use the difference of squares formula to factorize expressions with fractions?

A: Yes, you can use the difference of squares formula to factorize expressions with fractions. For example, if you have the expression $\frac{x^2}{4} - 1$ , you can factor it using the difference of squares formula as follows:

\frac{x^2}{4} - 1 = \frac{x^2}{4} - \frac{4}{4} = \frac{x^2 - 4}{4} = \frac{(x + 2)(x - 2)}{4}

Q: Can I use the difference of squares formula to factorize expressions with decimals?

A: Yes, you can use the difference of squares formula to factorize expressions with decimals. For example, if you have the expression $2.5^2 - 1.2^2$ , you can factor it using the difference of squares formula as follows:

2.5^2 - 1.2^2 = (2.5 + 1.2)(2.5 - 1.2) = 3.7 \cdot 1.3

Q: Can I use the difference of squares formula to factorize expressions with exponents?

A: Yes, you can use the difference of squares formula to factorize expressions with exponents. For example, if you have the expression $x^{10} - 1$ , you can factor it using the difference of squares formula as follows:

x^{10} - 1 = (x^5 + 1)(x^5 - 1) = (x^5 + 1)(x^4 + x^3 + x^2 + x + 1)(x - 1)

Q: Can I use the difference of squares formula to factorize expressions with radicals?

A: Yes, you can use the difference of squares formula to factorize expressions with radicals. For example, if you have the expression $\sqrt{2}^2 - 1$ , you can factor it using the difference of squares formula as follows:

\sqrt{2}^2 - 1 = (\sqrt{2} + 1)(\sqrt{2} - 1)

Q: Can I use the difference of squares formula to factorize expressions with complex numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with complex numbers. For example, if you have the expression $i^2 - 1$ , you can factor it using the difference of squares formula as follows:

i^2 - 1 = (i + 1)(i - 1)

Q: Can I use the difference of squares formula to factorize expressions with imaginary numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with imaginary numbers. For example, if you have the expression $i^4 - 1$ , you can factor it using the difference of squares formula as follows:

i^4 - 1 = (i^2 + 1)(i^2 - 1) = (i^2 + 1)(i + 1)(i - 1)

Q: Can I use the difference of squares formula to factorize expressions with mixed numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with mixed numbers. For example, if you have the expression $2\frac{1}{2}^2 - 1$ , you can factor it using the difference of squares formula as follows:

2\frac{1}{2}^2 - 1 = (2\frac{1}{2} + 1)(2\frac{1}{2} - 1) = \frac{5}{2} \cdot \frac{1}{2}

Q: Can I use the difference of squares formula to factorize expressions with negative exponents?

A: Yes, you can use the difference of squares formula to factorize expressions with negative exponents. For example, if you have the expression $x^{-2} - 1$ , you can factor it using the difference of squares formula as follows:

x^{-2} - 1 = (x^{-1} + 1)(x^{-1} - 1)

Q: Can I use the difference of squares formula to factorize expressions with fractional exponents?

A: Yes, you can use the difference of squares formula to factorize expressions with fractional exponents. For example, if you have the expression $x^{\frac{1}{2}} - 1$ , you can factor it using the difference of squares formula as follows:

x^{\frac{1}{2}} - 1 = (x^{\frac{1}{4}} + 1)(x^{\frac{1}{4}} - 1)

Q: Can I use the difference of squares formula to factorize expressions with irrational numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with irrational numbers. For example, if you have the expression $\sqrt{2}^2 - 1$ , you can factor it using the difference of squares formula as follows:

\sqrt{2}^2 - 1 = (\sqrt{2} + 1)(\sqrt{2} - 1)

Q: Can I use the difference of squares formula to factorize expressions with transcendental numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with transcendental numbers. For example, if you have the expression $e^2 - 1$ , you can factor it using the difference of squares formula as follows:

e^2 - 1 = (e + 1)(e - 1)

Q: Can I use the difference of squares formula to factorize expressions with complex transcendental numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with complex transcendental numbers. For example, if you have the expression $i^2 - 1$ , you can factor it using the difference of squares formula as follows:

i^2 - 1 = (i + 1)(i - 1)

Q: Can I use the difference of squares formula to factorize expressions with mixed transcendental numbers?

A: Yes, you can use the difference of squares formula to factorize expressions with mixed transcendental