Third degree polynomials are also known as cubic polynomials. Cubics have these characteristics: One to three roots. Two or zero extrema. Four points or pieces of information are required to define a cubic polynomial function.

A cubic equation is an algebraic equation of third-degree. The general form of a cubic function is: f (x) = ax3 + bx2 + cx1 + d. And the cubic equation has the form of ax3 + bx2 + cx + d = 0, where a, b and c are the coefficients and d is the constant.

Cubic Trinomials of the Form Ax^3 + Bx+^2 + Cx For example, the greatest common factor of the trinomial 3x^3 – 6x^2 – 9x is 3x, so the polynomial is equal to 3x times the trinomial x^2 – 2x -3, or 3x*(x^2 – 2x – 3). For example, the polynomial x^2 – 2x – 3 factors as (x – 3)(x + 1).

Degree of a Polynomial

zero

Types of Polynomials

• Monomial: An algebraic expression that contains only one non-zero term is known as a monomial.
• Binomial: An algebraic expression that contains two non zero terms is known as a binomial.
• Trinomial: An algebraic expression that contains three non-zero terms is known as the Trinomial.

A polynomial having its highest degree one is called a linear polynomial. For example, f(x) = x- 12, g(x) = 12 x , h(x) = -7x + 8 are linear polynomials. In general g(x) = ax + b , a ≠ 0 is a linear polynomial. For example, f (x) = 8×3 + 2×2 – 3x + 15, g(y) = y3 – 4y + 11 are cubic polynomials.

You call an expression with a single term a monomial, an expression with two terms is a binomial, and an expression with three terms is a trinomial. For example a polynomial with five terms is called a five-term polynomial.

Polynomial Definition. Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one term. In the polynomial, each expression in it is called a term. Suppose x2 + 5x + 2 is polynomial, then the expressions x2, 5x, and 2 are the terms of the polynomial.

Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. A polynomial having value zero (0) is called zero polynomial. The degree of a polynomial is the highest power of the variable x.

Degree of Polynomial

1. Linear Polynomial: Degree is 1. g. p(x)=8x – 2.
2. Quadratic Polynomial: Degree is 2. g. p(x)= 3×2 +8x – 2.
3. Cubic Polynomial: Degree is 3. g. p(x)=9×3 – 3×2 +8x – 2.

Factor Theorem. Factor Theorem. x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if x – a is a factor of p(x), then p(a) = 0, where a is any real number. This is an extension to remainder theorem where remainder is 0, i.e. p(a) = 0.

Using factor theorem, if x-1 is a factor of 2×4+3×2-5x+7, then by putting x=1, the given polynomial should equal to zero. Since the polynomial is not equal to zero, x-1 is not a factor of 2×4+3×2-5x+7.

In algebraic math, the factor theorem is a theorem that establishes a relationship between factors and zeros of a polynomial. Thus the factor theorem states that a polynomial has a factor if and only if: The polynomial x – M is a factor of the polynomial f(x) if and only if f (M) = 0.

Remainder Theorem is an approach of Euclidean division of polynomials. According to this theorem, if we divide a polynomial P(x) by a factor ( x – a); that isn’t essentially an element of the polynomial; you will find a smaller polynomial along with a remainder.

Definition of Remainder Theorem: Let p(x) be any polynomial of degree greater than or equal to 1 and let α be any real number. If p(x) is divided by the polynomial (x – α), then the remainder is p(α).

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial has a factor if and only if (i.e. is a root).

Basically, the remainder theorem links remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its zeros.

Answer: The Factor Theorem explain us that if the remainder f(r) = R = 0, then (x − r) happens to be a factor of f(x). The Factor Theorem is quite important because of its usefulness to find roots of polynomial equations.

Any time you divide by a number (being a potential root of the polynomial) and get a zero remainder in the synthetic division, this means that the number is indeed a root, and thus “x minus the number” is a factor.