To find the polynomial whose graph passes through the given points, we use Lagrange interpolation or solve a system of equations for a polynomial of degree 3, as four points determine a cubic polynomial.

General Form of a Cubic Polynomial

The general form of a cubic polynomial is:

$P(x) = ax^3 + bx^2 + cx + d$

Step 1: Set up equations for the given points

Using the coordinates of the points, substitute $x$ and $y$ into the polynomial equation $P(x) = y$:

1. For $(-2, -16)$: $-16 = a(-2)^3 + b(-2)^2 + c(-2) + d$ $-16 = -8a + 4b - 2c + d$

2. For $(0, 2)$: $2 = a(0)^3 + b(0)^2 + c(0) + d$ $2 = d$

3. For $(1, -1)$: $-1 = a(1)^3 + b(1)^2 + c(1) + d$ $-1 = a + b + c + d$

4. For $(2, -4)$: $-4 = a(2)^3 + b(2)^2 + c(2) + d$ $-4 = 8a + 4b + 2c + d$

Step 2: Substitute $d = 2$ into the equations

Using $d = 2$, the equations become:

1. $-16 = -8a + 4b - 2c + 2$: $-18 = -8a + 4b - 2c$

2. $-1 = a + b + c + 2$: $-3 = a + b + c$

3. $-4 = 8a + 4b + 2c + 2$: $-6 = 8a + 4b + 2c$

Step 3: Solve the system of equations

We now solve the system of three equations:

1. $-8a + 4b - 2c = -18$
2. $a + b + c = -3$
3. $8a + 4b + 2c = -6$

Simplify the equations:

1. Divide the first equation by 2: $-4a + 2b - c = -9$

2. Leave the second equation as is: $a + b + c = -3$

3. Divide the third equation by 2: $4a + 2b + c = -3$

Add equations (1) and (3):

$(-4a + 2b - c) + (4a + 2b + c) = -9 + (-3)$ $4b = -12 \implies b = -3$

Substitute $b = -3$ into equation (2):

$a - 3 + c = -3$ $a + c = 0 \implies c = -a$

Substitute $b = -3$ and $c = -a$ into equation (1):

$-4a + 2(-3) - (-a) = -9$ $-4a - 6 + a = -9$ $-3a = -3 \implies a = 1$

Find $c$:

$c = -a \implies c = -1$

Step 4: Write the polynomial

Now that we know $a = 1$, $b = -3$, $c = -1$, and $d = 2$, the polynomial is:

$P(x) = x^3 - 3x^2 - x + 2$

Final Answer:

$P(x) = x^3 - 3x^2 - x + 2$

Would you like to see the detailed graph or verification of the solution? Let me know!

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5 Follow-Up Questions:

1. How can Lagrange interpolation be used to derive the same polynomial?
2. Can we verify this polynomial by plugging in the given points?
3. What happens if we add an extra point to this problem?
4. How does the degree of a polynomial relate to the number of points it passes through?
5. What are the applications of polynomial interpolation in real-world problems?

Tip:

When solving systems of equations, always simplify each step to avoid algebraic errors, especially when substituting values!