SQA Higher Mathematics

## Polynomials A **polynomial** in $x$ is an expression made of powers of $x$ with whole-number indices, each multiplied by a constant called a coefficient, such as $4x^3 - x^2 + 7x - 5$. The highest power is its **degree**, so this one is a cubic, and the coefficient of that power is the **leading coefficient**. A polynomial of degree 4 is a quartic, and the constant term is the term with no $x$. A number $x = a$ for which $f(a) = 0$ is a **root** of $f(x) = 0$ and a **zero** of the function $f$. Finding roots is the same problem as finding linear factors, which is what the factor theorem says. ## The remainder theorem When a polynomial $f(x)$ is divided by $(x - h)$, the remainder is $f(h)$. The reason is that dividing gives $f(x) = (x - h)q(x) + r$ with $r$ a constant, and putting $x = h$ makes the first term vanish, leaving $f(h) = r$. So the remainder can be found without any division. To find the remainder when $f(x) = x^3 + 4x^2 - 2x + 5$ is divided by $(x + 2)$, put $x = -2$: $$ f(-2) = -8 + 16 + 4 + 5 = 17. $$ For a divisor such as $(2x - 1)$, the value to substitute is the one that makes the divisor zero, here $x = \frac{1}{2}$. ## The factor theorem $(x - h)$ is a factor of $f(x)$ exactly when $f(h) = 0$. This is the remainder theorem with remainder zero, and it works both ways: a zero of the remainder means a factor, and a factor means a root. In the same way $(ax - b)$ is a factor when $f\!\left(\frac{b}{a}\right) = 0$. The theorem is also used to find unknown…