The Polynomial Below Is A Perfect Square Trinomial Of The Form $A^2 - 2AB + B^2$.$9x^2 - 30x + 25$A. True B. False

Understanding Perfect Square Trinomials

A perfect square trinomial is a polynomial that can be expressed as the square of a binomial. It is of the form $(a-b)^2$ or $(a+b)^2$, where $a$ and $b$ are constants or variables. The general form of a perfect square trinomial is $A^2 - 2AB + B^2$, where $A$ and $B$ are constants or variables.

The Given Polynomial

The given polynomial is $9x^2 - 30x + 25$. To determine whether it is a perfect square trinomial, we need to check if it can be expressed in the form $(a-b)^2$ or $(a+b)^2$.

Breaking Down the Polynomial

Let's break down the given polynomial into its factors:

$$9x^2 - 30x + 25 = (3x)^2 - 2(3x)(5) + 5^2$$

We can see that the first term is the square of $3x$, the second term is $-2$ times the product of $3x$ and $5$, and the third term is the square of $5$.

Is it a Perfect Square Trinomial?

To determine whether the polynomial is a perfect square trinomial, we need to check if it can be expressed in the form $(a-b)^2$ or $(a+b)^2$. Let's try to factor the polynomial:

$$9x^2 - 30x + 25 = (3x - 5)^2$$

We can see that the polynomial can be expressed as the square of the binomial $(3x - 5)$. Therefore, it is a perfect square trinomial.

Conclusion

The given polynomial $9x^2 - 30x + 25$ is a perfect square trinomial of the form $(a-b)^2$. It can be expressed as the square of the binomial $(3x - 5)$. Therefore, the correct answer is:

A. True

Why is it Important to Identify Perfect Square Trinomials?

Identifying perfect square trinomials is important in algebra because it allows us to factor polynomials and solve equations more easily. Perfect square trinomials can be factored using the formula $(a-b)^2$ or $(a+b)^2$, which can be used to solve equations and inequalities.

Examples of Perfect Square Trinomials

Here are some examples of perfect square trinomials:

• $(x-3)^2 = x^2 - 6x + 9$
• $(x+4)^2 = x^2 + 8x + 16$
• $(2x-1)^2 = 4x^2 - 4x + 1$

How to Identify Perfect Square Trinomials

To identify perfect square trinomials, we need to check if the polynomial can be expressed in the form $(a-b)^2$ or $(a+b)^2$. We can do this by breaking down the polynomial into its factors and checking if it can be expressed as the square of a binomial.

Tips and Tricks

Here are some tips and tricks for identifying perfect square trinomials:

• Look for the form $(a-b)^2$ or $(a+b)^2$
• Check if the polynomial can be expressed as the square of a binomial
• Use the formula $(a-b)^2$ or $(a+b)^2$ to factor the polynomial

Conclusion

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Q: What is a perfect square trinomial?

A: A perfect square trinomial is a polynomial that can be expressed as the square of a binomial. It is of the form $(a-b)^2$ 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 check if the polynomial can be expressed in the form $(a-b)^2$ or $(a+b)^2$. You can do this by breaking down the polynomial into its factors and checking if it can be expressed as the square of a binomial.

Q: What is the general form of a perfect square trinomial?

A: The general form of a perfect square trinomial is $A^2 - 2AB + B^2$, where $A$ and $B$ are constants or variables.

Q: Can a perfect square trinomial be factored?

A: Yes, a perfect square trinomial can be factored using the formula $(a-b)^2$ or $(a+b)^2$. This can be used to solve equations and inequalities.

Q: What are some examples of perfect square trinomials?

A: Here are some examples of perfect square trinomials:

• $(x-3)^2 = x^2 - 6x + 9$
• $(x+4)^2 = x^2 + 8x + 16$
• $(2x-1)^2 = 4x^2 - 4x + 1$

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you need to use the formula $(a-b)^2$ or $(a+b)^2$. For example, if you have the perfect square trinomial $x^2 - 6x + 9$, you can factor it as $(x-3)^2$.

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

A: Here are some common mistakes to avoid when identifying perfect square trinomials:

• Not checking if the polynomial can be expressed as the square of a binomial
• Not using the formula $(a-b)^2$ or $(a+b)^2$ to factor the polynomial
• Not checking if the polynomial has the general form $A^2 - 2AB + B^2$

Q: How do I use perfect square trinomials to solve equations and inequalities?

A: Perfect square trinomials can be used to solve equations and inequalities by factoring the polynomial and then using the formula $(a-b)^2$ or $(a+b)^2$ to solve for the variable.

Q: What are some real-world applications of perfect square trinomials?

A: Perfect square trinomials have many real-world applications, including:

• Algebraic geometry
• Number theory
• Cryptography
• Optimization problems

Q: Can perfect square trinomials be used to solve quadratic equations?

A: Yes, perfect square trinomials can be used to solve quadratic equations. By factoring the polynomial and then using the formula $(a-b)^2$ or $(a+b)^2$ to solve for the variable, you can solve quadratic equations.

Q: What are some tips and tricks for identifying perfect square trinomials?

A: Here are some tips and tricks for identifying perfect square trinomials:

• Look for the form $(a-b)^2$ or $(a+b)^2$
• Check if the polynomial can be expressed as the square of a binomial
• Use the formula $(a-b)^2$ or $(a+b)^2$ to factor the polynomial

Conclusion

In conclusion, perfect square trinomials are an important concept in algebra that can be used to factor polynomials and solve equations and inequalities. By understanding how to identify and factor perfect square trinomials, you can solve a wide range of problems in algebra and beyond.