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