Find an equation of the cuble polynomial f(x)=a x^3+b x^2+c x+d that passes through the given po

Find an equation of the cuble polynomial f(x)=a x^3+b x^2+c...

Key Concepts

Polynomial Interpolation

Polynomial interpolation is the process of finding a polynomial that exactly passes through a set of provided data points. For a given set of n+1 distinct points, there exists a unique polynomial of degree at most n that interpolates those points, which is a fundamental concept in numerical analysis and data fitting.

System of Linear Equations

When determining the coefficients of a polynomial that must pass through specific points, each point generates an equation when its coordinates are substituted into the polynomial's general formula. This results in a system of linear equations in the unknown coefficients. Solving this system, using methods such as substitution, elimination, or matrix operations, is key to finding the unique polynomial that meets the interpolation conditions.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three expressed in the general form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a is non-zero. This type of polynomial has characteristics such as up to three real roots and potential inflection points, making it a versatile model for various types of behavior in mathematical analysis.

Transcript

00:01 In this problem, there is a given function expressed as f of x is equal to a xq plus px squared plus cx plus d.

00:12 And a given condition says that this specific function passes through the following point, pq, r, and s.

00:20 So they know that if a given function passes through a given point, then we know that plugging in the x and the y coordinates of that point to the function, we should, surely know that it will satisfy the equation.

00:35 Now the end goal for this problem is to identify what the value of a, b, c, and d, so that we can find out what is the actual coefficient of the variables of the variable x in this given function, f of x.

00:50 So for us to be able to do that, what we first want to do is to plug in the x and the y coordinates of this point pqr and s the given function.

01:01 So let's start.

01:02 Point p.

01:05 So plugging in point p to the function, we have 0 comma 4.

01:13 Take note that the first coordinate represents x and the second coordinate represents the f of x or the y.

01:21 Okay.

01:23 So plugging in the coordinates of p on the given function will have 4 is equal to a times 0 cube plus b times 0 squared, the c times 0 plus d.

01:39 So both a, b, c, and c are all going to be 0 because it's being multiplied to a zero value that being said will end up having d is equal to 4 for this one.

01:53 And this is the first variable we were able to calculate.

01:58 Now for q, plugging in the values of q to the function will have 2 is equal to a times 1 q.

02:08 Plus b times 1 squared the c times 1 plus d.

02:14 We'll have 2 is equal to a plus b plus c plus 4.

02:20 We know the value of 4.

02:22 We know the value of b it's equal to 4 so we can plug in 4d.

02:29 And this should give us a simplified equation as a plus b plus c is equal to negative 2.

02:36 And let's call it equation number 1.

02:41 Now, plugging in point r, which is negative 1 comma 10, will have 10 is equal to a times negative 1 cube plus b times negative 1 squared, plus c times negative 1, let me erase that.

02:58 It's a bit messy, plus d, which is equal to 4.

03:05 So we'll end up having 6 right here, negative a plus b minus, or let me.

03:15 Just try to write it this way let me erase that one and instead write negative a plus b minus c is equal to six and let's call it the equation number two now plugging in the last point point s it coordinates to negative two will have negative two is equal to a times two cube plus b times 2 squared, because c times 2, plus 4.

03:54 D is equal to 4.

03:57 Okay? so we'll end up having, excuse me, we'll end up having 8a plus 4b plus 2c is equal to negative 6.

04:14 All right, and let's call it equation number 2.

04:21 So we have now three equations with three unknown variables a b c this is now a system of linear equation and we have a bunch of method to solve this and i will try to solve this using elimination method so the first thing i wanted to eliminate is variable c eliminate c in equation one and two and how am i going to do that so i will simply just add equation one let me erase this one i will simply add a equation 1 with equation 2.

05:04 Note that equation 1 is a plus b plus c is equal to negative 2.

05:12 And equation number 2 is negative a plus b minus is equal to 6.

05:17 And take note by looking at it, it seems like if i will add this to equation, i can actually eliminate two variables.

05:24 So here, let me erase this a little bit.

05:27 And let's say eliminate a and c in equation 1 and so adding these two equations, we can cancel out a, cancel out c, and end up having 2b is equal to positive 4...

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