Find the function f(x)=a x^3+b x^2+ c x+d for which f(-2)=-10, f(-1)=3, f(1)=5, and f(3)=15

Name: Problem 78, Chapter 12: Algebra and Trigonometry
Uploaded: 2020-06-22T04:51:29-08:00
Duration: 11 min 19 s
Channel: Christine Anacker
Description: Find the function f(x)=a x^3+b x^2+ c x+d for which f(-2)=-10, f(-1)=3, f(1)=5, and f(3)=15

step 1 So we're doing this button in the curve given four functions. So we're going to create our 4, th

Find the function f(x)=a x^3+b x^2+ c x+d for which...

Key Concepts

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, generally written in the form ax³ + bx² + cx + d. Understanding cubic polynomials is essential for studying curves that can change direction up to two times and exhibit various interesting behaviors. This concept is fundamental in algebra and calculus, particularly in analyzing the shape and roots of curves.

Polynomial Interpolation

Polynomial interpolation is the process of determining a polynomial that exactly fits a set of given data points. In the context of a cubic polynomial, if four points are provided, one can uniquely determine the coefficients of the polynomial. This concept is widely used in numerical analysis and data fitting to construct functions that model a set of data.

System of Linear Equations

When determining the coefficients of a polynomial using given values, a system of linear equations is formed where each equation corresponds to the polynomial evaluated at a specific point. Solving these simultaneous equations is key to finding the unknown coefficients. This method forms a bridge between linear algebra and polynomial function analysis.

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