Completely factor the polynomial, if possible. \[ 9 x^{2}-12 x+4 \] AnswerHow to enter your answer (opens in new window) 2 Points Keypad Keyboard Shortcuts Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answer is used. \( \square \) Not a Special Factorization

This polynomial is a quadratic in the form of \(ax^2 + bx + c\) where \(a = 9\), \(b = -12\), and \(c = 4\). To factor it, we can first look for a common factor. Notice that each term can be divided by 1, but let’s also check for perfect squares because \(9x^2\) and \(4\) are perfect squares themselves. This polynomial can be rewritten as: \[ (3x - 2)^2 \] This means \(9x^2 - 12x + 4 = (3x - 2)(3x - 2)\). To confirm this, we can use the FOIL method. When you expand \((3x - 2)^2\) you get \(3x \cdot 3x - 2 \cdot 3x - 2 \cdot 3x + 2 \cdot 2\) which simplifies down to \(9x^2 - 12x + 4\). So, we’ve successfully factored the polynomial completely! As a fun fact, it’s important to remember that not all quadratic equations can be factored neatly like this one; in some cases, you might encounter roots that aren’t integers or simple fractions. One popular mistake when factoring quadratics is to overlook the possibility of completing the square or using the quadratic formula when factoring isn't straightforward. Always double-check by multiplying your factors back together to ensure they match the original polynomial!