Finding Linear, Quadratic and Cubic polynomials

Linear, Quadratic, and Cubic polynomials from Table

A polynomial in X is an algebraic expression of the form f(x) = a0 + a1x + a2x2 + a3 x3 +..........+ an xn, where a1, a2, a3....an are real numbers and all the indexes of 'x' are non-negative integers. Polynomial is derived from the words "poly" and "nomial," which together mean "many terms." Constants, variables, and exponents can all be found in a polynomial.

The highest degree of a polynomial's exponent(variable) with a non-zero coefficient is its degree. The term "degree" means "power" in this context. Let's look at different degrees of polynomials in this article.

Degree of a Polynomial

The degree of a polynomial is the highest degree exponent term.

All you have to do to find the degree is find the largest exponent in the given polynomial.

For instance, consider the following equation:

x3 + 2x2 + 4x + 3 = f(x). The equation has a degree of three. The degree of the polynomial is defined as the highest power of the variable in the polynomial.

A polynomial of degree 2 is f(x) = 7x2 - 3x + 12.

f(x) = an xn + an-1 xn-1 + an-2xn-2 +...................+ a1 x + a0 where a0, a1, a2......an are constants and a 0

We have the following names for the degree of polynomial based on the degree of polynomial.

Constant Polynomial

A constant polynomial is a polynomial with the highest degree

zero. There are no variables in it, only constants.

For example, constant polynomials are f(x) = 6, g(x) = -22, h(y) = 5/2,

and so on. F(x) = c is a constant polynomial in general.

The zero polynomial is the constant polynomial 0 or f(x) = 0.

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Linear Polynomials

A linear polynomial is a polynomial with the highest degree

number.

Linear polynomials include f(x) = x-12, g(x) = 12 x, and h(x) = -7x + 8.g(x) = axe + b is a linear polynomial in general.

A polynomial having its highest degree 2 is known as a quadratic

polynomial.

For example, f (x) = 2x2 - 3x + 15, g(y) = 3/2 y2 - 4y + 11 are quadratic polynomials.

In general g(x) = ax2+ bx + c, a ≠ 0 is a quadratic polynomial.

Cubic Polynomial

A polynomial having its highest degree 3 is known as a Cubic

polynomial.

For example, f (x) = 8x3 + 2x2- 3x + 15, g(y) =  y3 - 4y + 11 are cubic polynomials.

In general g(x) = ax3 + bx2 + cx + d, a ≠ 0 is a quadratic polynomial.

The general linear, quadratic, and cubic functions are represented in the following three different tables. These tables will be used to locate functions.

Using finite-difference tables, find the rule for each of the following sequences:

A 20, 19, 18, 17,16, ………  B 5, 11, 19, 29, 41, ………. C 6, 26, 64, 126, 218, 346, ………

A-   Solution

 X F(X) 1st Difference 1 20 -1 2 19 -1 3 18 -1 4 17 -1 5 16

The function is a Linear of the form f(X) = ax + b. From the difference table for the quadratic

a = -1 and therefore a = -1 (using column 1st Diff 2)

Finally, a + b = 20 = -1 + b = 20 and therefore b = 21 (using column F(X))

f (X) = -1X + 21

B-   Solution

X      f(X)    ∆1    ∆2

The function is a quadratic of the form f (X) = aX2 + bx + c. From the difference table for the quadratic 2a = 2 and therefore a = 1 (using column ∆2)

Also, 3a + b = 6 and therefore b = 3 (using column ∆1)

Finally, a + b + c = 5 and therefore c = 1 (using column f (X))

Therefore, F(X) = X2 + 3X + 1

C-   Solution

X        f(X)         ∆1        ∆2         ∆3

Therefore f(n) = ax3 + bx2 + cx + d

6a = 6                          a = 1 (from column ∆1)

12a + 2b = 18              b = 3 (from column ∆2)

7a + 3b + c = 20         c = 4 (from column ∆3)

a + b + c + d = 6         d = 2 (from column f (x))

f (x) = x^3 + 3x^2 + 4x − 2