Which Products Result In A Difference Of Squares? Select Three Options.A. \[$(x-y)(y-x)\$\]B. \[$(6-y)(6-y)\$\]C. \[$(3+xz)(-3+xz)\$\]D. \[$\left(y^2-xy\right)\left(y^2+xy\right)\$\]E.

In algebra, a difference of squares is a mathematical expression that can be factored into the product of two binomials. It is a fundamental concept in mathematics, and understanding how to identify and factor these expressions is crucial for solving various mathematical problems. In this article, we will explore which products result in a difference of squares.

What is a Difference of Squares?

A difference of squares is a mathematical expression of the form:

$$a^2 - b^2$$

where $a$ and $b$ are any real numbers. This expression can be factored into the product of two binomials as:

$$(a+b)(a-b)$$

For example, the expression $9x^2 - 4y^2$ is a difference of squares, and it can be factored as:

$$(3x+2y)(3x-2y)$$

Option A: {(x-y)(y-x)$}$

At first glance, option A may seem like a difference of squares. However, upon closer inspection, we can see that it is actually a product of two binomials that are the same, but with opposite signs. This is not a difference of squares, but rather a product of two identical binomials.

To see why, let's expand the product:

$$(x-y)(y-x) = x^2 - xy - xy + y^2 = x^2 - 2xy + y^2$$

As we can see, this is not a difference of squares, but rather a quadratic expression in the form of $x^2 - 2xy + y^2$.

Option B: {(6-y)(6-y)$}$

Option B is another product of two binomials that are the same, but with opposite signs. This is not a difference of squares, but rather a product of two identical binomials.

To see why, let's expand the product:

$$(6-y)(6-y) = 36 - 6y - 6y + y^2 = 36 - 12y + y^2$$

As we can see, this is not a difference of squares, but rather a quadratic expression in the form of $36 - 12y + y^2$.

Option C: {(3+xz)(-3+xz)$}$

Option C is a product of two binomials that are the same, but with opposite signs. This is not a difference of squares, but rather a product of two identical binomials.

To see why, let's expand the product:

$$(3+xz)(-3+xz) = -9 + 3xz + 3xz - x^2z^2 = -9 + 6xz - x^2z^2$$

As we can see, this is not a difference of squares, but rather a quadratic expression in the form of $-9 + 6xz - x^2z^2$.

Option D: {\left(y^{2-xy\right)\left(y}2+xy\right)$}$

Option D is a product of two binomials that are the same, but with opposite signs. This is actually a difference of squares, and it can be factored as:

$$\left(y^2-xy\right)\left(y^2+xy\right) = (y^2)^2 - (xy)^2 = (y^2 - xy)(y^2 + xy)$$

As we can see, this is a difference of squares, and it can be factored into the product of two binomials.

Conclusion

In conclusion, only option D is a difference of squares. The other options are not differences of squares, but rather products of two identical binomials. Understanding how to identify and factor differences of squares is crucial for solving various mathematical problems, and it is an essential concept in algebra.

Key Takeaways

• A difference of squares is a mathematical expression of the form $a^2 - b^2$.
• A difference of squares can be factored into the product of two binomials as $(a+b)(a-b)$.
• Only option D is a difference of squares.
• The other options are not differences of squares, but rather products of two identical binomials.

Final Thoughts

In our previous article, we explored which products result in a difference of squares. In this article, we will answer some frequently asked questions about difference of squares.

Q: What is a difference of squares?

A: A difference of squares is a mathematical expression of the form $a^2 - b^2$, where $a$ and $b$ are any real numbers.

Q: How do I identify a difference of squares?

A: To identify a difference of squares, look for an expression in the form $a^2 - b^2$. If the expression can be written in this form, it is a difference of squares.

Q: How do I factor a difference of squares?

A: To factor a difference of squares, use the formula $(a+b)(a-b)$. This will give you the factored form of the expression.

Q: Can I factor a difference of squares with variables?

A: Yes, you can factor a difference of squares with variables. For example, the expression $x^2 - 4y^2$ is a difference of squares, and it can be factored as $(x+2y)(x-2y)$.

Q: Can I factor a difference of squares with fractions?

A: Yes, you can factor a difference of squares with fractions. For example, the expression $\frac{1}{4}x^2 - \frac{1}{9}y^2$ is a difference of squares, and it can be factored as $\frac{1}{36}(3x+2y)(3x-2y)$.

Q: Can I factor a difference of squares with negative numbers?

A: Yes, you can factor a difference of squares with negative numbers. For example, the expression $-x^2 + 4y^2$ is a difference of squares, and it can be factored as $-(x+2y)(x-2y)$.

Q: Can I factor a difference of squares with imaginary numbers?

A: Yes, you can factor a difference of squares with imaginary numbers. For example, the expression $x^2 + 4y^2$ is a difference of squares, and it can be factored as $(x+2iy)(x-2iy)$.

Q: Can I factor a difference of squares with complex numbers?

A: Yes, you can factor a difference of squares with complex numbers. For example, the expression $x^2 + 4y^2$ is a difference of squares, and it can be factored as $(x+2iy)(x-2iy)$.

Q: Can I factor a difference of squares with exponents?

A: Yes, you can factor a difference of squares with exponents. For example, the expression $x^4 - 4y^4$ is a difference of squares, and it can be factored as $(x^2+2y^2)(x^2-2y^2)$.

Q: Can I factor a difference of squares with radicals?

A: Yes, you can factor a difference of squares with radicals. For example, the expression $x^2 - 4y^2$ is a difference of squares, and it can be factored as $(x+2y)(x-2y)$.

Q: Can I factor a difference of squares with trigonometric functions?

A: Yes, you can factor a difference of squares with trigonometric functions. For example, the expression $\sin^2x - \cos^2x$ is a difference of squares, and it can be factored as $-\sin x\cos x(\sin x+\cos x)$.

Q: Can I factor a difference of squares with logarithmic functions?

A: Yes, you can factor a difference of squares with logarithmic functions. For example, the expression $\log^2x - \log^2y$ is a difference of squares, and it can be factored as $(\log x - \log y)(\log x + \log y)$.

Conclusion

In this article, we answered some frequently asked questions about difference of squares. We saw that difference of squares can be factored into the product of two binomials, and we learned how to identify and factor these expressions. Understanding how to identify and factor differences of squares is crucial for solving various mathematical problems, and it is an essential concept in algebra.

Key Takeaways

• A difference of squares is a mathematical expression of the form $a^2 - b^2$.
• A difference of squares can be factored into the product of two binomials as $(a+b)(a-b)$.
• Difference of squares can be factored with variables, fractions, negative numbers, imaginary numbers, complex numbers, exponents, radicals, trigonometric functions, and logarithmic functions.

Final Thoughts

In this article, we explored some frequently asked questions about difference of squares. We saw that difference of squares can be factored into the product of two binomials, and we learned how to identify and factor these expressions. Understanding how to identify and factor differences of squares is crucial for solving various mathematical problems, and it is an essential concept in algebra.