- A perfect square trinomial follows the pattern \((x+k)^2 = x^2 + 2kx + k^2\), which requires the constant term \(c\) to equal \(\left(\frac{b}{2}\right)^2\).
- Checking each option:
- \(x^2 - 6x + 9\): \(\left(\frac{-6}{2}\right)^2 = 9\), which matches.
- \(x^2 - 2x + 4\): \(\left(\frac{-2}{2}\right)^2 = 1
eq 4\).
- \(x^2 + 5x + 10\): \(\left(\frac{5}{2}\right)^2 = 6.25
eq 10\).
- \(x^2 + 4x + 16\): \(\left(\frac{4}{2}\right)^2 = 4
eq 16\).
- The correct polynomial is \(\boxed{x^2 - 6x + 9}\).
Explanation
- Understanding Perfect Square Trinomials
A perfect square trinomial is a quadratic expression of the form \(x^2 + 2kx + k^2\), which factors as \((x + k)^2\). To determine if a trinomial \(x^2 + bx + c\) is a perfect square, we check if the constant term \(c\) is equal to the square of half the coefficient of \(x\), that is, \(c = \left(\frac{b}{2}\right)^2\). If this condition holds, the expression can be written as \(\left(x + \frac{b}{2}\right)^2\).
- Testing Each Option
We examine each of the four given options to see which satisfies the condition \(c = \left(\frac{b}{2}\right)^2\):
- For \(x^2 - 6x + 9\): Here \(b = -6\) and \(c = 9\). Calculating \(\left(\frac{b}{2}\right)^2 = \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9\). Since \(9 = 9\), this is a perfect square trinomial, specifically \((x - 3)^2\).
- For \(x^2 - 2x + 4\): Here \(b = -2\) and \(c = 4\). Calculating \(\left(\frac{b}{2}\right)^2 = \left(\frac{-2}{2}\right)^2 = (-1)^2 = 1\). Since \(1
eq 4\), this is not a perfect square trinomial.
- For \(x^2 + 5x + 10\): Here \(b = 5\) and \(c = 10\). Calculating \(\left(\frac{b}{2}\right)^2 = \left(\frac{5}{2}\right)^2 = 2.5^2 = 6.25\). Since \(6.25
eq 10\), this is not a perfect square trinomial.
- For \(x^2 + 4x + 16\): Here \(b = 4\) and \(c = 16\). Calculating \(\left(\frac{b}{2}\right)^2 = \left(\frac{4}{2}\right)^2 = 2^2 = 4\). Since \(4
eq 16\), this is not a perfect square trinomial.
- Conclusion
Based on our analysis, the only expression that satisfies the definition of a perfect square trinomial is \(x^2 - 6x + 9\). Thus, the correct choice is the first one. We conclude that \(\boxed{x^2 - 6x + 9}\) is the polynomial representing a perfect square model.
Examples
In architecture or landscaping, if you have a square plot of land and you know the area can be represented by a mathematical expression, you might use perfect square trinomials to determine the side lengths of the plots. For example, if the area of a square garden is given by \(x^2 - 6x + 9\) square meters, you can factor this to \((x-3)^2\), telling you that each side of the square garden is \((x-3)\) meters long. This is a practical application of identifying perfect square structures in real-world dimensions.