The topic of this problem sheet will be polynomials. Before we dive into the examples, let’s recap a few key concepts.
A polynomial in \(x\) is an expression formed by adding or subtracting monomials, which are terms of the form \(a x^n\), where \(a\) is a number called a coefficient, and \(n\) is a whole number (non-negative integer). Here, \(x\) is a variable that may represent a number. The degree of a polynomial \(f\), written as \(\deg(f)\) is the highest power of \(x\) appearing in the polynomial. For example: \(\deg(x^3+x^2+x)=3\). Thus, we write the general form of a polynomial of degree \(n\) as
\[p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots + a_1x+a_0,\] where \(a_n\) must be non-zero (otherwise the degree would be \(n-1\)). We can perform several familiar operations on polynomials, which you may have seen before:
Addition and subtraction: We add or subtract polynomials by looking at each power of \(x\) and adding or subtracting the corresponding coefficients. Generally: \[(a_nx^n+a_{n-1}x^{n-1}+\cdots + a_1+a_0)+(b_nx^n+b_{n-1}x^{n-1}+\cdots + b_1+b_0)=(a_n+b_n)x^n + (a_{n-1}+b_{n-1})x^{n-1}+\cdots + (a_1+b_1)x+(a_0+b_0).\]
For example, if \[f(x) = x^4 + 3x - 1 \quad \text{and} \quad g(x) = x^3 + 2x + 5,\] then \(f(x) - g(x) = x^4 - x^3 + x - 6\).
Multiplication: We use the distributive property, which means that every term in the first polynomial is multiplied by every term in the second polynomial. Generally:
\[(a_nx^n+\cdots a_1x+a_0)(b_mx^m +\cdots + b_1 x + b_0) = a_nb_m x^{n+m}+\cdots + a_ib_j x^{i+j}+\cdots + a_0b_0,\] where \(1\leq i \leq n\) and \(1\leq j \leq m\). For example, if \[f(x) = x^2 + x + 1 \quad \text{and} \quad g(x) = x - 1,\] then \(f(x) g(x) = (x^2 + x + 1)(x - 1) = x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1.\)
Let’s now present the examples. They have some very important techniques, so read them carefully before attempting the problems.