Algebra of Polynomials
Algebra of Polynomials
If we consider two polynomials, \( P(x) \) and \( Q(x) \), defined respectively as \( P(x) = a_0x^n + a_1x^{n-1} + ... + a_n \), and \( Q(x) = b_0x^m + b_1x^{m-1} + ... + b_m \), where \( \delta(P) \) is the degree of \( P \) which is \( n \), and \( \delta(Q) \) is the degree of \( Q \) which is \( m \), various operations such as addition, subtraction, multiplication, and division can be performed on these polynomials following algebraic rules.
Addition of Polynomials
The sum of two polynomials \( P(x) \) and \( Q(x) \) is defined as \( (P + Q)(x) = P(x) + Q(x) \), meaning that the value of \( P + Q \) at any point \( x \) is the sum of the values of \( P \) at \( x \) and \( Q \) at \( x \).
The degree of the resulting polynomial \( P + Q \) is determined by the higher degree of the two polynomials \( P \) and \( Q \):
However, if \( \delta(P) = \delta(Q) \), the resulting degree may be less than \( \delta(P) \) or \( \delta(Q) \) if the leading terms cancel each other out.
For example, consider the polynomials \( P(x) = x^3 + 2x^2 - x + 1 \) and \( Q(x) = -x^3 - 2x^2 + 6x - 2 \). The sum \( (P + Q)(x) \) simplifies to \( -7x + 3 \). Here, both \( P \) and \( Q \) have a degree of 3, but \( \delta(P + Q) \) is 1 because the terms containing \( x^3 \) and \( x^2 \) cancel each other out.
Multiplication of Polynomials
The product of two polynomials \( P(x) \) and \( Q(x) \) is denoted as \( PQ(x) = P(x)Q(x) \). This means that the value of \( PQ \) at any point \( x \) is the product of the values of \( P \) at \( x \) and \( Q \) at \( x \).
The degree of the product polynomial \( PQ \) is the sum of the degrees of \( P \) and \( Q \):
This property is straightforward to demonstrate.
For example, take \( P(x) = x^3 + 2x - 1 \) and \( Q(x) = 4x^2 + 2x - 1 \). The product \( PQ(x) \) is given by multiplying each term in \( P(x) \) by each term in \( Q(x) \), leading to:
The resulting polynomial after multiplying and combining like terms would be:
[ PQ(x) = 4x^5 + \ldots ] (with additional terms resulting from the multiplication process).
Here, \( \delta(P) = 3 \) and \( \delta(Q) = 2 \), so according to the rule, \( \delta(PQ) = 3 + 2 = 5 \), which aligns with the leading term \( 4x^5 \) in the expanded product.
Composition of Polynomials
The composition of two polynomials \( P \) and \( Q \) is denoted by \( P(Q(x)) \), which means that the polynomial \( Q(x) \) is substituted for every occurrence of \( x \) in polynomial \( P(x) \). In other words, \( P(Q(x)) = a_0(Q(x))^n + a_1(Q(x))^{n-1} + \ldots + a_n \), where the \( x \) in the expression of \( P(x) \) is completely replaced by \( Q(x) \).
To illustrate, let's take \( P(x) = 4x^2 - 2x + 3 \) and \( Q(x) = x^2 - x \). Then the composition \( P(Q(x)) \) is:
When computing this, you combine like terms to get the resulting polynomial.
The degree \( \delta \) of the composition \( P(Q(x)) \) is the product of the degrees of \( P \) and \( Q \):
For instance, if \( P \) and \( Q \) both have a degree of 2, then \( P(Q(x)) \) would have a degree of 4, assuming there's no cancellation of the highest degree terms.
If \( P(x) \) has the same degree as \( P(Q(x)) \), it implies that the degree of \( Q(x) \) must be 1, which means \( Q(x) \) is a linear polynomial. This is because the composition of a polynomial with a linear polynomial does not change the degree of the original polynomial.