Identifying Perfect Square Trinomials

Jul 14, 2025 by ADMIN 38 views

Determining which expression results in a perfect square trinomial requires a solid understanding of what a perfect square trinomial is and how it's formed. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In simpler terms, it's the result of squaring a binomial. Mathematically, this can be represented as:

$(a + b)^2 = a^2 + 2ab + b^2$ or $(a - b)^2 = a^2 - 2ab + b^2$

Where a and b are any algebraic terms. Let's break down the key components:

$a^2$ and $b^2$ : These are the squares of the first and second terms of the binomial, respectively.
$2ab$ : This is twice the product of the first and second terms of the binomial. The sign of this term will determine whether the original binomial was a sum or a difference.

Now, let's examine the given expressions to identify which one fits this pattern. We will expand each expression and see if it matches the form of a perfect square trinomial. It's important to pay close attention to the signs and coefficients, as these are critical in determining the correct result. Our goal is to find an expression that, when expanded, can be written in the form $a^2 ± 2ab + b^2$ . This process involves applying the distributive property (often referred to as FOIL – First, Outer, Inner, Last) to multiply the binomials and then simplifying the resulting expression. By carefully analyzing each expansion, we can pinpoint the expression that yields a perfect square trinomial. The correct expression will not only have the squared terms ( $a^2$ and $b^2$ ) but also the crucial middle term ( $2ab$ ) with the correct sign and coefficient. This methodical approach ensures that we accurately identify the expression that perfectly fits the definition of a perfect square trinomial.

Analyzing the Given Expressions

Let's analyze each expression to determine which one results in a perfect square trinomial:

(3x - 5)(3x - 5) This expression represents the square of the binomial (3x - 5). Expanding this using the distributive property (FOIL method) gives us:

$(3x - 5)(3x - 5) = (3x)(3x) + (3x)(-5) + (-5)(3x) + (-5)(-5)$ $= 9x^2 - 15x - 15x + 25$ $= 9x^2 - 30x + 25$

This result, $9x^2 - 30x + 25$ , fits the pattern of a perfect square trinomial. Here, $a = 3x$ and $b = 5$ . We can see that $a^2 = (3x)^2 = 9x^2$ , $b^2 = (5)^2 = 25$ , and $2ab = 2(3x)(5) = 30x$ . The expression matches the form $a^2 - 2ab + b^2$ , confirming it as a perfect square trinomial. This expression is indeed a perfect square trinomial because it is the result of squaring the binomial (3x - 5). The expanded form clearly shows the squared terms ( $9x^2$ and 25) and the middle term (-30x), which is twice the product of 3x and -5. The signs are consistent with the pattern of a perfect square trinomial derived from the square of a difference. Therefore, this expression stands out as a clear example of a perfect square trinomial among the given options.
(3x - 5)(5 - 3x) Expanding this expression gives us:

$(3x - 5)(5 - 3x) = (3x)(5) + (3x)(-3x) + (-5)(5) + (-5)(-3x)$ $= 15x - 9x^2 - 25 + 15x$ $= -9x^2 + 30x - 25$

This is a trinomial, but it's not a perfect square. Although it has squared terms and a middle term, the signs do not align with the pattern of a perfect square trinomial. Specifically, the leading term (-9x^2) is negative, which is not characteristic of a perfect square trinomial derived from the square of a binomial. This expression, when expanded, results in a trinomial with a negative leading coefficient, making it distinct from a perfect square trinomial. Perfect square trinomials, by definition, arise from squaring a binomial, which always results in a positive leading coefficient. The presence of -9x^2 immediately disqualifies this expression. Therefore, a closer examination of the signs and coefficients reveals that this trinomial does not fit the required pattern for a perfect square trinomial.
(3x - 5)(3x + 5) Expanding this expression gives us:

$(3x - 5)(3x + 5) = (3x)(3x) + (3x)(5) + (-5)(3x) + (-5)(5)$ $= 9x^2 + 15x - 15x - 25$ $= 9x^2 - 25$

This result is a difference of squares, not a trinomial. The middle terms cancel each other out, leaving only the squared terms with a subtraction sign between them. This outcome is characteristic of the product of the sum and difference of two terms, a pattern distinctly different from a perfect square trinomial. While the expression does involve squared terms, the absence of a middle term means it cannot be classified as a perfect square trinomial. The expansion clearly demonstrates the cancellation of the cross-product terms, resulting in a binomial rather than a trinomial. Therefore, this expression exemplifies the difference of squares pattern, a contrast to the perfect square trinomial pattern we are looking for.
(3x - 5)(-3x - 5) Expanding this expression gives us:

$(3x - 5)(-3x - 5) = (3x)(-3x) + (3x)(-5) + (-5)(-3x) + (-5)(-5)$ $= -9x^2 - 15x + 15x + 25$ $= -9x^2 + 25$

This result is also a difference, but with a negative leading term. Like the previous example, it's not a perfect square trinomial. The expression simplifies to a binomial with a negative coefficient for the squared term, which is not consistent with the form of a perfect square trinomial. Perfect square trinomials, when derived from the square of a binomial, always have a positive leading coefficient. The negative sign on the $x^2$ term indicates that this expression does not fit the perfect square trinomial pattern. Therefore, this outcome further emphasizes the distinction between a difference of squares with a negative leading term and a perfect square trinomial, which requires a specific pattern of terms and signs.

Conclusion

In conclusion, among the given expressions, (3x - 5)(3x - 5) is the only one that results in a perfect square trinomial. It expands to $9x^2 - 30x + 25$ , which fits the form $a^2 - 2ab + b^2$ . This detailed analysis, involving expansion and pattern matching, highlights the importance of understanding the properties of perfect square trinomials and how they are derived from the square of a binomial. The correct expression, (3x - 5)(3x - 5), perfectly illustrates this concept, showcasing the characteristic squared terms and the crucial middle term that defines a perfect square trinomial. This exercise underscores the significance of algebraic manipulation and pattern recognition in identifying and working with various types of polynomial expressions.