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.
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. 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. 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.