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 nonnegative 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
nonzero 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 + an1 xn1 + an2xn2 +...................+ 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.
Equation Formatter
Linear Polynomials
A linear polynomial is a polynomial with the highest degree
number.
Linear polynomials include f(x) = x12, g(x) = 12 x, and h(x) = 7x + 8.g(x) = axe + b is a linear polynomial in general.
Quadratic Polynomial
A polynomial having its highest degree 2 is known as a quadratic
polynomial.
For example, f (x) = 2x^{2 } 3x + 15, g(y) = 3/2 y^{2 } 4y + 11 are quadratic polynomials.
In general g(x) = ax^{2}+ 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) = 8x^{3 }+ 2x^{2} 3x + 15, g(y) = y^{3 } 4y + 11 are cubic polynomials.
In general g(x) = ax^{3 }+ bx^{2 }+ 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 finitedifference 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

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 1^{st} 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
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