Which Products Result In A Perfect Square Trinomial? Select All That Apply.A. ( − X + 9 ) ( − X − 9 (-x+9)(-x-9 ( − X + 9 ) ( − X − 9 ]B. ( X Y + X ) ( X Y + X (x Y+x)(x Y+x ( X Y + X ) ( X Y + X ]C. ( 2 X − 3 ) ( − 3 + 2 X (2 X-3)(-3+2 X ( 2 X − 3 ) ( − 3 + 2 X ]D. \left(16-x^2\right)\left(x^2-16\right ]E. $\left(4 Y^2+25\right)\left(25+4

Introduction

In algebra, a perfect square trinomial is a polynomial expression that can be factored into the square of a binomial. It is a quadratic expression that can be written in the form of $(a+b)^2$ or $(a-b)^2$, where $a$ and $b$ are constants or variables. In this article, we will explore which products result in a perfect square trinomial.

What is a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It is a polynomial expression that can be written in the form of $(a+b)^2$ or $(a-b)^2$, where $a$ and $b$ are constants or variables. For example, $(x+3)^2$ is a perfect square trinomial because it can be factored into $(x+3)(x+3)$.

How to Identify a Perfect Square Trinomial

To identify a perfect square trinomial, we need to look for the following characteristics:

• The expression must be a quadratic expression, meaning it must have a degree of 2.
• The expression must be in the form of $(a+b)^2$ or $(a-b)^2$, where $a$ and $b$ are constants or variables.
• The expression must be able to be factored into the square of a binomial.

Which Products Result in a Perfect Square Trinomial?

Now that we have discussed what a perfect square trinomial is and how to identify one, let's examine the products listed in the question.

A. $(-x+9)(-x-9)$

To determine if this product results in a perfect square trinomial, we need to multiply the two binomials together.

$(-x+9)(-x-9) = x^2 - 9x - 9x + 81$

Simplifying the expression, we get:

$x^2 - 18x + 81$

This expression is not a perfect square trinomial because it cannot be factored into the square of a binomial.

B. $(x y+x)(x y+x)$

To determine if this product results in a perfect square trinomial, we need to multiply the two binomials together.

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

Simplifying the expression, we get:

$x^2 y^2 + 2x^2 y + x^2$

This expression is not a perfect square trinomial because it cannot be factored into the square of a binomial.

C. $(2 x-3)(-3+2 x)$

To determine if this product results in a perfect square trinomial, we need to multiply the two binomials together.

$(2 x-3)(-3+2 x) = -6 + 4 x^2 - 6 x + 6 x$

Simplifying the expression, we get:

$4 x^2 - 6 x - 6 + 6 x$

This expression is not a perfect square trinomial because it cannot be factored into the square of a binomial.

D. $\left(16-x^2\right)\left(x^2-16\right)$

To determine if this product results in a perfect square trinomial, we need to multiply the two binomials together.

$\left(16-x^2\right)\left(x^2-16\right) = x^4 - 16 x^2 - 16 x^2 + 256$

Simplifying the expression, we get:

$x^4 - 32 x^2 + 256$

This expression is a perfect square trinomial because it can be factored into the square of a binomial.

E. $\left(4 y^2+25\right)\left(25+4 y^2\right)$

To determine if this product results in a perfect square trinomial, we need to multiply the two binomials together.

$\left(4 y^2+25\right)\left(25+4 y^2\right) = 16 y^4 + 100 y^2 + 100 y^2 + 625$

Simplifying the expression, we get:

$16 y^4 + 200 y^2 + 625$

This expression is a perfect square trinomial because it can be factored into the square of a binomial.

Conclusion

In conclusion, the products that result in a perfect square trinomial are:

• $\left(16-x^2\right)\left(x^2-16\right)$
• $\left(4 y^2+25\right)\left(25+4 y^2\right)$

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about perfect square trinomials.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It is a polynomial expression that can be written in the form of $(a+b)^2$ or $(a-b)^2$, where $a$ and $b$ are constants or variables.

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, you need to look for the following characteristics:

• The expression must be a quadratic expression, meaning it must have a degree of 2.
• The expression must be in the form of $(a+b)^2$ or $(a-b)^2$, where $a$ and $b$ are constants or variables.
• The expression must be able to be factored into the square of a binomial.

Q: What are some examples of perfect square trinomials?

A: Some examples of perfect square trinomials include:

• $(x+3)^2$
• $(x-2)^2$
• $(y+4)^2$
• $(y-1)^2$

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you need to follow these steps:

1. Identify the binomial that is being squared.
2. Write the binomial in the form of $(a+b)$ or $(a-b)$.
3. Square the binomial by multiplying it by itself.
4. Simplify the expression to get the final factored form.

Q: What are some common mistakes to avoid when working with perfect square trinomials?

A: Some common mistakes to avoid when working with perfect square trinomials include:

• Not recognizing that an expression is a perfect square trinomial.
• Not factoring the expression correctly.
• Not simplifying the expression to get the final factored form.

Q: How do I use perfect square trinomials in real-world applications?

A: Perfect square trinomials are used in a variety of real-world applications, including:

• Algebra: Perfect square trinomials are used to solve quadratic equations and to factor quadratic expressions.
• Geometry: Perfect square trinomials are used to find the area and perimeter of shapes.
• Physics: Perfect square trinomials are used to describe the motion of objects.

Q: What are some advanced topics related to perfect square trinomials?

A: Some advanced topics related to perfect square trinomials include:

• Quadratic equations: Perfect square trinomials are used to solve quadratic equations.
• Factoring: Perfect square trinomials are used to factor quadratic expressions.
• Algebraic geometry: Perfect square trinomials are used to describe the geometry of shapes.

Conclusion

In conclusion, perfect square trinomials are an important concept in algebra and are used in a variety of real-world applications. By understanding how to identify and factor perfect square trinomials, you can solve quadratic equations and factor quadratic expressions.