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
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.
To find the polynomial that fits the given points, we can use a process called interpolation. This involves using the given points to set up a system of equations and solving for the coefficients of the polynomial.
The general form of a polynomial is y = ax^n + bx^(n-1) + ... + cx + d, where a, b, c, and d are coefficients, and n is the degree of the polynomial.
Using the given points, we can substitute the x and y values into the polynomial equation to get a system of equations. Let's set up the equations using the points:
For the point (-4, 89), we have:
89 = a(-4)^n + b(-4)^(n-1) + c(-4) + d
For the point (-3, 7), we have:
7 = a(-3)^n + b(-3)^(n-1) + c(-3) + d
For the point (-1, -1), we have:
-1 = a(-1)^n + b(-1)^(n-1) + c(-1) + d
For the point (1, -1), we have:
-1 = a(1)^n + b(1)^(n-1) + c(1) + d
For the point (4, 329), we have:
329 = a(4)^n + b(4)^(n-1) + c(4) + d
Now, we can solve this system of equations to find the values of a, b, c, and d. However, since the question does not provide a specific value for n (degree of the polynomial), we cannot determine the exact polynomial.
You can solve the system of equations using various methods like substitution, elimination, or matrix methods to find the coefficients. Once you have the coefficients, you can write the polynomial equation in standard form.
Based on the provided answer choices, you can select the polynomial equation:
y = x^4 - 2x^3 + 3x^2 + 2x - 1.
Please note that the exact polynomial equation might differ depending on the degree of the polynomial and the specific coefficients obtained from solving the system of equations.
To find the polynomial that passes through the given points, we can use the method of interpolation.
The points given are: (–4, 89), (–3, 7), (–1, –1), (1, –1), (4, 329).
To find the polynomial, we need to find the degree of the polynomial. Since we have 5 points, we need a polynomial of degree 4.
We can set up a system of equations using the coordinates of the points:
For the point (–4, 89):
89 = a(-4)^4 + b(-4)^3 + c(-4)^2 + d(-4) + e
For the point (–3, 7):
7 = a(-3)^4 + b(-3)^3 + c(-3)^2 + d(-3) + e
For the point (–1, –1):
-1 = a(-1)^4 + b(-1)^3 + c(-1)^2 + d(-1) + e
For the point (1, –1):
-1 = a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e
For the point (4, 329):
329 = a(4)^4 + b(4)^3 + c(4)^2 + d(4) + e
Solving this system of equations will give us the values of a, b, c, d, and e, which are the coefficients of the polynomial.
Using a mathematical software or calculator to solve this system of equations, we find that the coefficients are:
a = 1, b = 2, c = -3, d = -2, e = 1
Therefore, the polynomial that passes through the given points is:
y = x^4 + 2x^3 - 3x^2 - 2x + 1.
So the correct choice is:
y = x4 + 2x3 – 3x2 – 2x + 1