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In this section, we explore various types of polynomials defined by their terms and degrees, along with their significance as mathematical expressions. The concept of zeros and the Factor Theorem are also explained.
In this section, we delve into the structure and characteristics of polynomials. A polynomial in one variable is expressed as a sum of terms, each comprising a coefficient and a variable raised to a certain power. The classification of polynomials includes monomials (one term), binomials (two terms), and trinomials (three terms), along with specific types based on their degrees: linear (degree one), quadratic (degree two), and cubic (degree three). The concept of zeros, or roots, of polynomials is crucial, as a real number 'a' is a zero if substituting it into the polynomial results in zero. The Factor Theorem further connects the roots of a polynomial with its factors. This section concludes with specific polynomial identities, illustrating the expansion of binomials and the sum of cubes.
Polynomial: A mathematical expression containing variables of non-negative integer powers.
Monomial, Binomial, Trinomial: Different categories of polynomials based on the number of terms.
Degree: The highest exponent in a polynomial, indicating its type.
Zero of a Polynomial: Values that make the polynomial equal to zero.
Factor Theorem: A principle connecting factors with their roots.
Polynomials are fun to see, monomials, binomials, come join me!
Imagine climbing a hill (the highest degree). Each step (the terms) counts, but together they show the way.
For degrees: 'L, Q, C' means Linear, Quadratic, Cubic.
If p(x) = 2x^3 + 3x^2 - x + 5, then it is a cubic polynomial of degree 3.
For p(x) = x^2 - 4, the zero is a = 2 because p(2) = 0.
Term: Polynomial
Definition: An algebraic expression in one variable that consists of terms of the form anxn + an–1xn–1 + ... + a2x2 + a1x + a0.
An algebraic expression in one variable that consists of terms of the form anxn + an–1xn–1 + ... + a2x2 + a1x + a0.
Term: Monomial
Definition: A polynomial with only one term.
A polynomial with only one term.
Term: Binomial
Definition: A polynomial with two terms.
A polynomial with two terms.
Term: Trinomial
Definition: A polynomial with three terms.
A polynomial with three terms.
Term: Degree
Definition: The highest exponent of the variable in a polynomial.
The highest exponent of the variable in a polynomial.
Term: Zero of a Polynomial
Definition: A value 'a' such that p(a) = 0.
A value 'a' such that p(a) = 0.
Term: Factor Theorem
Definition: States that if x – a is a factor of p(x), then p(a) = 0.
States that if x – a is a factor of p(x), then p(a) = 0.