91Ó°ÊÓ

Understanding perfect square trinomials is crucial for factoring expressions like the one given in the exercise. A perfect square trinomial is a special type of polynomial that can be written in the form \(a^2 - 2ab + b^2\). Notice that this form results when you square a binomial \(a - b\). For example, consider our trinomial \(16x^2 - 40x + 25\) from the exercise. Here’s how it fits the pattern:
\[ (4x)^2 = 16x^2 \] and \[ 5^2 = 25 \]. So the terms 16x^2 and 25 are perfect squares of 4x and 5, respectively. To verify, check the middle term \(-2ab\):
Let \a = 4x\ and \b = 5\. The middle term is \(-2(4x)(5) = -40x\), which matches the given trinomial. Therefore, we conclude it’s a perfect square trinomial.

For perfect square trinomials, the factored form is straightforward. It simplifies the trinomial into a product of two identical binomials. Given our example \(16x^2 - 40x + 25\), we determined earlier that it matches \(a^2 - 2ab + b^2\). Here, \[ a = 4x \] and \[ b = 5 \]. Therefore, the trinomial can be factored as \((4x - 5)^2\). Writing it out clearly:
\[(4x - 5)^2\]
means \[(4x - 5)(4x - 5)\]. This shows that the trinomial can be expressed as two multiplied binomials, simplifying your algebraic work and solving equations that involve such trinomials.

Factoring trinomials involves algorithmic steps and a bit of practice to recognize patterns. Let’s break down the algebra steps for solving our exercise:

• Step 1: Identify if the trinomial fits the perfect square form \(a^2 - 2ab + b^2\).
• For \(16x^2 - 40x + 25\), we identify \[ a = 4x \] and \[ b = 5 \] that satisfy \ (4x)^2 = 16x^2 and \ 5^2 = 25 \.
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• Step 2: Verify the middle term to ensure it fits the form \(-2ab\). Here, \(-2(4x)(5) = -40x\).
• Step 3: Write the trinomial in its factored form \((a - b)^2\). In this case, \((4x - 5)^2\).

Following these structured steps, any student can factor trinomials effectively.