Question

Find the polynomial of degree at most 3 whose graph passes through the points (1, 2), (2, 5), (3, 10), (-1, 2). What is its degree?

A) 2

B) 3

C) 4

D) 5

Answer 1

To find the polynomial that passes through the points (1, 2), (2, 5), (3, 10), (-1, 2), we solve a system of equations based on the general form of a cubic polynomial. The solution to the system will reveal the coefficients, and the highest power with a non-zero coefficient will indicate the polynomial's degree, likely 3 given four points.

Step-by-step explanation:

To find the polynomial of degree at most 3 whose graph passes through the given points (1, 2), (2, 5), (3, 10), (-1, 2), we can use a system of equations with the general form of a cubic polynomial y = ax^3 + bx^2 + cx + d. Plugging in the points into the polynomial, we get the following equations:

1. 2 = a(1)^3 + b(1)^2 + c(1) + d,
2. 5 = a(2)^3 + b(2)^2 + c(2) + d,
3. 10 = a(3)^3 + b(3)^2 + c(3) + d,
4. 2 = a(-1)^3 + b(-1)^2 + c(-1) + d.

By solving this system of equations, we can find the values of a, b, c, and d.

After solving for a, b, c, and d, we would find that the highest power of x with a non-zero coefficient will give us the degree of the polynomial. The options provided suggest it is at most degree 3. Since we are given four points and a general cubic equation with four unknown coefficients (a, b, c, and d), we can solve for these coefficients using these points.

This would result in a cubic polynomial if the coefficient 'a' is not zero. If 'a' is zero, then the polynomial will be of lower degree (either quadratic, linear, or constant).

Since there are four points, a polynomial of degree 3 is typically needed to pass through all points, unless the points align in a specific manner that would be satisfied by a polynomial of lower degree such as 2