Find a polynomial p of degree 3 so that
p(4) = 5,
p(−1) = 5,
p(−3) = −37,
p(2) = −7,
then use your polynomial to approximate p(1).
p(x) = 0
p(1) = 0
there is only one cubic which passes through all four of those points:
p(x) = x^3-3x^2-4x+5
no need to approximate p(1), since you can see that it is -1.
As for p(x) = 0, you'll have to approximate that, using a graph, or some other numeric tool you have studied, such as: ??
To find a polynomial that satisfies the given conditions, we can use interpolation methods. Since we have four data points, we can construct a polynomial of degree 3.
Let's work through the steps to find the polynomial p(x):
Step 1: Set up the polynomial equation
A general polynomial of degree 3 can be expressed as:
p(x) = ax^3 + bx^2 + cx + d
Step 2: Substitute the given values
Using the given conditions, we can form a system of equations:
p(4) = 5,
p(-1) = 5,
p(-3) = -37,
p(2) = -7
Substituting the values into the polynomial equation, we get:
64a + 16b + 4c + d = 5, (Equation 1)
-a + b - c + d = 5, (Equation 2)
-27a + 9b - 3c + d = -37, (Equation 3)
8a + 4b + 2c + d = -7. (Equation 4)
Step 3: Solve the system of equations
To solve this system of equations, we can use any method, such as substitution or elimination. To simplify the process, we'll use the Gauss-Jordan elimination method.
Writing the system of equations as an augmented matrix, we have:
[ 64 16 4 1 | 5]
[-1 1 -1 1 | 5]
[-27 9 -3 1 |-37]
[ 8 4 2 1 |-7]
Applying row operations, we get:
[ 1 1/4 1/16 1/64 | 5/64]
[ 0 1 -1 2 | 11]
[ 0 0 1 -1 | -7]
[ 0 0 0 1 | -5]
Step 4: Extract the coefficients
From the reduced row-echelon form, we have:
a = -5, b = -25/4, c = 9/16, d = 5/64
Therefore, the polynomial p(x) is:
p(x) = -5x^3 - (25/4)x^2 + (9/16)x + (5/64)
Step 5: Find p(1)
To approximate p(1), we substitute x = 1 into the polynomial:
p(1) = -5(1)^3 - (25/4)(1)^2 + (9/16)(1) + (5/64)
= -5 - 25/4 + 9/16 + 5/64
= -20/4 - 25/4 + 9/16 + 5/64
= (-80 - 100 + 18 + 5)/64
= -157/64
Therefore, p(1) ≈ -157/64.
Note: The given conditions uniquely determine a polynomial of degree 3 that satisfies all the conditions.