How to Determine if a Polynomial is a Perfect Square Trinomial

Oct 13, 2025 by miodeya.com 26 views

Introduction

In algebra, a perfect square trinomial is a polynomial that can be expressed as the square of a binomial. It is a trinomial that can be factored into the square of a binomial expression. In this article, we will determine which of the given polynomials is a perfect square trinomial.

What is a Perfect Square Trinomial?

A perfect square trinomial is a polynomial that can be expressed as the square of a binomial. It is a trinomial that can be factored into the square of a binomial expression. The general form of a perfect square trinomial is:

(a + b)^2 = a^2 + 2ab + b^2**

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

How to Determine if a Polynomial is a Perfect Square Trinomial

To determine if a polynomial is a perfect square trinomial, we need to check if it can be factored into the square of a binomial. We can do this by checking if the polynomial can be written in the form of a^2 + 2ab + b^2 or a^2 - 2ab + b^2.

Method 1: Check if the Polynomial can be Factored into the Square of a Binomial

We can check if a polynomial can be factored into the square of a binomial by trying to factor it. If the polynomial can be factored into the square of a binomial, then it is a perfect square trinomial.

Method 2: Check if the Polynomial satisfies the Perfect Square Trinomial Formula

We can also check if a polynomial satisfies the perfect square trinomial formula by checking if it can be written in the form of a^2 + 2ab + b^2 or a^2 - 2ab + b^2.

Example 1: Determine if $4x^2 - 12x + 9$ is a Perfect Square Trinomial

Let's try to determine if $4x^2 - 12x + 9$ is a perfect square trinomial.

We can start by trying to factor the polynomial:

$4x^2 - 12x + 9 = (2x - 3)^2$

Since the polynomial can be factored into the square of a binomial, it is a perfect square trinomial.

Example 2: Determine if $16x^2 + 24x - 9$ is a Perfect Square Trinomial

Let's try to determine if $16x^2 + 24x - 9$ is a perfect square trinomial.

We can start by trying to factor the polynomial:

$16x^2 + 24x - 9 = (4x + 3)(4x - 3)$

Since the polynomial cannot be factored into the square of a binomial, it is not a perfect square trinomial.

Example 3: Determine if $4a^2 - 10a + 25$ is a Perfect Square Trinomial

Let's try to determine if $4a^2 - 10a + 25$ is a perfect square trinomial.

We can start by trying to factor the polynomial:

$4a^2 - 10a + 25 = (2a - 5)^2$

Since the polynomial can be factored into the square of a binomial, it is a perfect square trinomial.

Example 4: Determine if $36b^2 - 24b - 16$ is a Perfect Square Trinomial

Let's try to determine if $36b^2 - 24b - 16$ is a perfect square trinomial.

We can start by trying to factor the polynomial:

$36b^2 - 24b - 16 = (6b - 4)(6b + 4)$

Since the polynomial cannot be factored into the square of a binomial, it is not a perfect square trinomial.

Conclusion

In conclusion, we have determined which of the given polynomials is a perfect square trinomial. The polynomials $4x^2 - 12x + 9$ and $4a^2 - 10a + 25$ are perfect square trinomials, while the polynomials $16x^2 + 24x - 9$ and $36b^2 - 24b - 16$ are not perfect square trinomials.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Final Answer

The final answer is:

Introduction

In our previous article, we determined which of the given polynomials is a perfect square trinomial. In this article, we will answer some frequently asked questions related to perfect square trinomials.

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 a trinomial that can be factored into the square of a binomial expression.

Q: How do I determine if a polynomial is a perfect square trinomial?

A: To determine if a polynomial is a perfect square trinomial, you can try to factor it into the square of a binomial. If the polynomial can be factored into the square of a binomial, then it is a perfect square trinomial.

Q: What are the general forms of a perfect square trinomial?

A: The general forms of a perfect square trinomial are:

(a + b)^2 = a^2 + 2ab + b^2**

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

Q: How do I check if a polynomial satisfies the perfect square trinomial formula?

A: To check if a polynomial satisfies the perfect square trinomial formula, you can try to write it in the form of a^2 + 2ab + b^2 or a^2 - 2ab + b^2. If the polynomial can be written in one of these forms, then it is a perfect square trinomial.

Q: Can a polynomial be a perfect square trinomial if it has a negative leading coefficient?

A: Yes, a polynomial can be a perfect square trinomial even if it has a negative leading coefficient. For example, the polynomial -x^2 + 4x - 4 can be factored into the square of a binomial: -(x - 2)^2.

Q: Can a polynomial be a perfect square trinomial if it has a variable with an even exponent?

A: Yes, a polynomial can be a perfect square trinomial even if it has a variable with an even exponent. For example, the polynomial x^4 + 4x^2 + 4 can be factored into the square of a binomial: (x^2 + 2)^2.

Q: Can a polynomial be a perfect square trinomial if it has a variable with a fractional exponent?

A: No, a polynomial cannot be a perfect square trinomial if it has a variable with a fractional exponent. For example, the polynomial x^2 + 2x^(1/2) + 1 cannot be factored into the square of a binomial.

Q: Can a polynomial be a perfect square trinomial if it has a variable with a negative exponent?

A: No, a polynomial cannot be a perfect square trinomial if it has a variable with a negative exponent. For example, the polynomial x^(-2) + 2x^(-1) + 1 cannot be factored into the square of a binomial.

Conclusion

In conclusion, we have answered some frequently asked questions related to perfect square trinomials. We hope that this article has been helpful in understanding the concept of perfect square trinomials.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Final Answer

The final answer is:

A. $4x^2 - 12x + 9$ C. $4a^2 - 10a + 25$