Given the following data: x: (-2, 1, 2), y: (28, 4, 16). Let \(P_2(x)\) be the quadratic interpolating polynomial passing through the above three points. Find the value of \(P_2(0)\).
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When asked to find the value of an interpolating polynomial at \(x=0\), you are essentially looking for the constant term (\(c\) in \(ax^2+bx+c\)). Setting up and solving the system of equations for \(c\) is often the most direct method.
Step 1: Assume polynomial form. $P_2(x) = ax^2 + bx + c$. We need $P_2(0) = c$. Step 2: Set up equations from data. 1) $4a - 2b + c = 28$ 2) $a + b + c = 4$ 3) $4a + 2b + c = 16$ Step 3: Solve for c. Add (1) and (3): $8a + 2c = 44 \implies 4a + c = 22$. Subtract (1) from (3): $4b = -12 \implies b = -3$. Substitute $b=-3$ in (2): $a - 3 + c = 4 \implies a + c = 7$. Now we have: $4a + c = 22$ and $a + c = 7$. Subtracting the second from the first: $3a = 15 \implies a = 5$. Finally: $5 + c = 7 \implies c = 2$.