Questions: Factor the trinomial (16 z^2-56 z+49) (16 x^2-56 x+49=(A x-B)^2) where (A) is (square) and (B) is (square).

Solution Steps

The given trinomial is \(16z^{2} - 56z + 49\). We are asked to express it in the form \((Az - B)^{2}\). This suggests that the trinomial is a perfect square.

Expanding \((Az - B)^{2}\) gives: \[ (Az - B)^{2} = A^{2}z^{2} - 2ABz + B^{2} \] We can compare this with the given trinomial \(16z^{2} - 56z + 49\).

By comparing the coefficients, we get: \[ A^{2} = 16 \quad \text{and} \quad B^{2} = 49 \] Solving for \(A\) and \(B\): \[ A = \sqrt{16} = 4 \quad \text{and} \quad B = \sqrt{49} = 7 \] Next, we check the middle term: \[ -2AB = -56 \quad \Rightarrow \quad -2 \cdot 4 \cdot 7 = -56 \] This confirms that the values of \(A\) and \(B\) are correct.

Final Answer