Question: Factor completely. (Hint: Look for a perfect square trinomial.) \[ 25z^2 - 40z + 16 =…

Factor completely. (Hint: Look for a perfect square trinomial.)

\[ 25z^2 - 40z + 16 = \boxed{\phantom{0}} \]

Solution

To factor the quadratic expression \( 25z^2 - 40z + 16 \) completely, we will look for two numbers whose product is \( 25 \times 16 = 400 \) and whose sum is \(-40\). The numbers \(-20\) and \(-20\) satisfy these conditions. Rewrite the middle term: \[ 25z^2 - 20z - 20z + 16 \] Group the terms: \[ (25z^2 - 20z) + (-20z + 16) \] Factor each group: \[ 5z(5z - 4) - 4(5z - 4) \] Factor out the common factor \((5z - 4)\): \[ (5z - 4)(5z - 4) \] Thus, the complete factorization is: \[ (5z - 4)^2 \]