Question

What polynomial has a graph that passes through the given points?

(–4, 89), (–3, 7), (–1, –1), (1, –1), (4, 329)

(1 point)
Responses

y = 2x3 – 3x2 – 2x + 1
y = 2 x 3 – 3 x 2 – 2 x + 1

y = 1x4 – 2x3 – 3x2 + 2x + 1
y = 1 x 4 – 2 x 3 – 3 x 2 + 2 x + 1

y = x4 – 2x3 + 3x2 + 2x – 1
y = x 4 – 2 x 3 + 3 x 2 + 2 x – 1

y = x4 + 2x3 – 3x2 – 2x + 1

Answer 1

To determine the polynomial that passes through the given points, we can use the method of interpolation.

The given points are:
(–4, 89), (–3, 7), (–1, –1), (1, –1), (4, 329)

To find the polynomial, we need to calculate the coefficients of the polynomial equation.

Using the given points, we can set up a system of equations:

89 = a(-4)^4 + b(-4)^3 + c(-4)^2 + d(-4) + e
7 = a(-3)^4 + b(-3)^3 + c(-3)^2 + d(-3) + e
-1 = a(-1)^4 + b(-1)^3 + c(-1)^2 + d(-1) + e
-1 = a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e
329 = a(4)^4 + b(4)^3 + c(4)^2 + d(4) + e

Simplifying the equations, we get:

89 = 256a - 64b + 16c - 4d + e
7 = 81a - 27b + 9c - 3d + e
-1 = a - b + c - d + e
-1 = a + b + c + d + e
329 = 256a + 64b + 16c + 4d + e

Therefore, the coefficients are:
a = 1, b = 2, c = -3, d = -2, e = 1

The polynomial that passes through the given points is:
y = x^4 + 2x^3 - 3x^2 - 2x + 1

Therefore, the correct option is y = x^4 + 2x^3 - 3x^2 - 2x + 1.