Polynomial Algebra of Real Numbers
Summary:
In this class, we will explore polynomial algebra, its definition, properties, and applications. Polynomials are a fundamental part of mathematics and have broad applications in various disciplines.
LEARNING OBJECTIVES
By the end of this class, the student will be able to:
1. Define and understand polynomials and their properties.
2. Identify the degree and coefficients of a polynomial.
3. Perform algebraic operations with polynomials and apply their properties in mathematical contexts.
CONTENT INDEX:
1. Polynomial Algebra: Definitions
2. Types of Polynomials
3. Polynomial Algebra: Operations
4. Factorization and Division of Polynomials
1. Polynomial Algebra: Definitions
To understand Polynomial Algebra, we must first know what polynomials are. Polynomials are algebraic functions. If x is a real variable, then the function P(x) is called a polynomial if it can be written in the form:
\displaystyle P(x)= \sum_{i=0}^n a_i x^i= a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots + a_nx^n,
where n is a non-negative integer, and all a_i, with i\in\{1,2,3,\cdots,n\}, are real coefficients. If there is a k such that a_k\neq 0 and, when k\lt i, it follows that a_i=0, then such a value of k is called the degree of the polynomial. In other words, the degree of a polynomial is the highest power that accompanies a non-zero coefficient.
2. Types of Polynomials
Polynomials are classified according to their degree; therefore, when mentioning a polynomial, it is almost always stated that it is a polynomial of degree k, where k is the highest power of x that accompanies the non-zero coefficient of such a polynomial.
2.1. Constant Polynomials
This is the family that includes all zero-degree polynomials and the null polynomial. We say that a polynomial is of degree zero if it can be written in the form P(x)=c, with c\neq 0. On the other hand, the null polynomial is of the form P(x) = 0, and for this, a degree is not defined.
3. Polynomial Algebra: Operations
Polynomials inherit all their properties from the algebra of real numbers. Distributive and associative properties are especially relevant.
3.1. Addition and Subtraction
If P and Q are two polynomials of degree n and m, respectively, with
m=n+k and 0\leq k,
then it follows that:
\begin{array}{rl} \displaystyle P(x) \pm Q(x) &=\displaystyle \sum_{i=0}^n a_i x^i \pm \sum_{i=0}^m b_i x^i \\ \\ &\displaystyle = \sum_{i=0}^n a_i x^i \pm \left( \sum_{i=0}^n b_i x^i + \sum_{i=n+1}^{n+k} b_i x^i \right) \\ \\ &\displaystyle = \sum_{i=0}^n (a_i \pm b_i) x^i + \sum_{i=n+1}^m b_i x^i \end{array}
That is, the coefficients that accompany equal powers of x are added or subtracted, as appropriate.
EXAMPLE:
If P(x) = 3+5x+2x^2 and Q(x) = 6x-3x^2 +23x^5, then:
P(x) + Q(x) = \cdots \\ = (3+5x+2x^2) + (6x-3x^2 +23x^5) \\ = 3 + (5+6)x + (2-3)x^2 + 23x^5 \\ = 3 + 11x - x^2 + 23x^5
P(x) - Q(x) = \cdots \\ = (3+5x+2x^2) - (6x-3x^2 +23x^5) \\ = 3 + (5-6)x + (2+3)x^2 - 23x^5 \\ = 3 - x + 5x^2 - 23x^5
3.2. Multiplication
In the same context as for the addition and subtraction of polynomials, the product of polynomials will be developed as follows:
First, we distinguish scalar multiplication. If c \in \mathbb{R}, then we have:
\displaystyle c P(x) = c \sum_{i=0}^n a_i x^i =\sum_{i=0}^n c a_i x^i
And then we have the multiplication of polynomials:
\begin{array}{rl} \displaystyle P(x) Q(x) &\displaystyle = \left( \sum_{i=0}^n a_i x^i \right) \left(\sum_{j=0}^m b_j x^j\right) \\ \\ &=\displaystyle \left[\sum_{j=0}^m \left( \sum_{i=0}^n a_i x^i \right) b_j x^j\right] \\ \\ &=\displaystyle \sum_{j=0}^m \left( \sum_{i=0}^n a_ib_j x^{i+j} \right) \\ \\ &=\displaystyle \sum_{i,j=0}^{n,m} a_ib_j x^{i+j} \end{array}
This is what we would summarize as “the sum of the products of all with all.”
EXAMPLE:
If P(x) = 4x+ 2x^2-x^4 and Q(x) = 5 - x + x^2-7x^3, then:
P(x)Q(x) =\cdots \\ {} \\= (4x+ 2x^2-x^4)(5 - x + x^2-7x^3) \\ {} \\ = 4x(5 - x + x^2-7x^3) \\ + 2x^2 (5 - x + x^2-7x^3) \\ - x^4 (5 - x + x^2-7x^3) \\ {} \\ = 20x - 4x^2 + 4x^3 - 28x^4 \\ + 10x^2 - 2x^3 + 2x^4 - 14x^5 \\ -5x^4 + x^5 - x^6 + 7x^7 \\ {} \\ = 20x + 6x^2 + 2x^3 - 31x^4 - 13x^5 - x^6 + 7x^7
4. Factorization and Division of Polynomials
When we multiply two polynomials, we transform two simple polynomials into a more complex one (of higher degree). When we factorize a polynomial, we follow the inverse process: we transform a complex polynomial into the product of two or more lower-degree polynomials.
To factorize a polynomial P(x), it is necessary to find the values of x that nullify the polynomial; if such values exist, then the polynomial is factorable. Talking about existence is straightforward, but finding them is a different story. We will review this topic in more detail when we study the factorizations of quadratic and (2n) quadratic polynomials.
4.1. Notable Products
There are, however, cases where factorization is easily obtained,
as in the case of notable products. Some of these results are as follows:
x^2 - y^2 = (x-y)(x+y)
(x\pm y)^2 = x^2 \pm 2xy + y^2
(x \pm y)^3 = x^3 \pm 3x^2y + 3xy^2 \pm y^3
x^3-y^3=(x-y)(x^2+xy+y^2)
x^3+y^3=(x+y)(x^2-xy+y^2)
4.2. The Division Algorithm
Just as multiplying integers gives us composite numbers, and division through the division algorithm allows us to factorize when the remainder is zero, the same happens with polynomials. Explaining the division algorithm “in text” can be a bit complicated; it’s much easier to understand by directly seeing how it’s done and in which cases the algorithm leads to factorization. To achieve this, we will review some examples.
EXAMPLE: Calculate P(x):Q(x) for the following cases:
- P(x)=2 x^3 + x^2 - 2 x - 1, Q(x)=x-1 [SOLUTION]
- P(x)=x^4+2x^3-x+1, Q(x)=x^2-4 [SOLUTION]
- P(x)=3 x^4 - 2 x^3 - x^2 - 4 x + 1, Q(x)=x^2+x+1 [SOLUTION]
- P(x)=x^7+5x^4+5x^2-3x+1, Q(x)=x^3-2x^2+1 [SOLUTION]