Functions
Polynomial functions and the tools to master them — division, Ruffini's rule, the remainder theorem, factorization, inequalities — then operations on functions and rational functions.
- A quadratic \(ax^2 + bx + c = 0\) has roots \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- The domain is the set of allowed inputs; the zeros are where \(f(x)=0\).
5.1 Cubic and quartic functions
A polynomial function of degree \(n\) has the form \(P(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0\), with \(a_n \neq 0\). Degree 3 is cubic; degree 4 is quartic.
A polynomial of degree \(n\) has at most \(n\) real zeros and at most \(n-1\) turning points.
5.2 Polynomial division and Ruffini's rule
For polynomials \(P(x)\) and \(D(x)\neq 0\), there exist unique \(Q(x)\) (quotient) and \(R(x)\) (remainder) with
\[ P(x) = D(x)\,Q(x) + R(x), \qquad \deg R < \deg D. \]Ruffini's rule is a fast scheme for dividing by a linear factor \((x - a)\). Bring down the leading coefficient, multiply by \(a\), add to the next coefficient, and repeat.
Divide \(P(x) = x^3 - 6x^2 + 11x - 6\) by \((x - 1)\):
| 1 | −6 | 11 | −6 | |
|---|---|---|---|---|
| \(a=1\) | 1 | −5 | 6 | |
| 1 | −5 | 6 | 0 |
Quotient \(x^2 - 5x + 6\), remainder \(0\). So \(P(x) = (x-1)(x^2 - 5x + 6)\).
5.3 The remainder theorem
The remainder of dividing \(P(x)\) by \((x - a)\) equals \(P(a)\). Consequently,
\[ (x - a)\ \text{is a factor of}\ P(x) \iff P(a) = 0. \]Is \((x-2)\) a factor of \(P(x) = x^3 - 6x^2 + 11x - 6\)? \(P(2) = 8 - 24 + 22 - 6 = 0\), so yes.
5.4 Multiplicity of a root
A root \(a\) has multiplicity \(m\) if \((x-a)^m\) divides \(P(x)\) but \((x-a)^{m+1}\) does not. At a root of even multiplicity the graph touches the \(x\)-axis; at odd multiplicity it crosses.
5.5 Factorization into linear and quadratic factors
Combining the previous tools, a polynomial can be written as a product of linear factors (real roots) and irreducible quadratic factors.
From 5.2, \(x^3 - 6x^2 + 11x - 6 = (x-1)(x^2 - 5x + 6) = (x-1)(x-2)(x-3)\). A classic identity from the book cover: \(x^3 - 1 = (x-1)(x^2 + x + 1)\).
5.6 Polynomial equations and inequalities of degree > 2
To solve \(P(x) > 0\) (or \(< 0\)): factor \(P\), mark its zeros on a number line, and study the sign on each interval (a sign chart).
Solve \((x-1)(x-2)(x-3) > 0\). Zeros at \(1, 2, 3\) split the line into four intervals; testing signs gives positive on \(]1,2[\) and \(]3, +\infty[\). Solution: \(x \in\ ]1,2[\ \cup\ ]3, +\infty[\).
5.7 Operations with functions
The domain of a combination is the intersection of the domains (excluding zeros of the denominator for a quotient).
5.8 Rational functions
A rational function is a quotient of polynomials \(f(x) = \dfrac{P(x)}{Q(x)}\), defined where \(Q(x)\neq 0\). The basic model is \(f(x) = \dfrac{1}{x}\) (a hyperbola).
Vertical asymptote \(x = h\); horizontal asymptote \(y = k\). A useful derivative fact (see Unit 6): \(\left(\tfrac{1}{x}\right)' = -\dfrac{1}{x^2}\), as printed on the book cover.
For \(f(x) = \dfrac{2}{x-3} + 1\): domain \(x \neq 3\), vertical asymptote \(x = 3\), horizontal asymptote \(y = 1\).
Practice problems
- Use Ruffini to divide \(2x^3 + 3x^2 - 11x - 6\) by \((x-2)\).
Show solution
Coefficients \(2, 3, -11, -6\) with \(a=2\): bring 2; \(2\cdot2+3=7\); \(7\cdot2-11=3\); \(3\cdot2-6=0\). Quotient \(2x^2 + 7x + 3\), remainder 0. - Find all real roots of \(x^3 - 6x^2 + 11x - 6\).
Show solution
\((x-1)(x-2)(x-3)=0 \Rightarrow x = 1, 2, 3\). - Solve \(x^2(x - 4) \le 0\).
Show solution
\(x^2 \ge 0\); the product is \(\le 0\) when \(x - 4 \le 0\), i.e. \(x \le 4\) (and \(x=0\) gives 0). Solution: \(x \in\ ]{-\infty}, 4]\). - State the domain and asymptotes of \(f(x) = \dfrac{1}{x + 2}\).
Show solution
Domain \(x \neq -2\); vertical asymptote \(x = -2\); horizontal asymptote \(y = 0\).
Summary
- Division algorithm: \(P = DQ + R\); Ruffini divides quickly by \((x-a)\).
- Remainder theorem: remainder \(= P(a)\); factor theorem: \(P(a)=0 \iff (x-a)\mid P\).
- Factor fully, then use a sign chart to solve polynomial inequalities.
- Rational functions: exclude zeros of the denominator; identify vertical/horizontal asymptotes.
- Find the remainder when \(x^4 - 1\) is divided by \((x - 1)\).
- Factor \(x^3 + 1\).