Revise Polynomial functions, factor and remainder theorems for IB Mathematics: Analysis and Approaches (2.7)

A polynomial function is $P(x)=a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$, where the coefficients are real numbers and the powers are non-negative integers. Two numbers control the overall shape. The degree $n$ is the highest power; it caps the number of real roots at $n$ and the number of turning points at $n-1$. The leading coefficient $a_n$ scales the dominant term. Together they fix the end behaviour — what the graph does for large $|x|$ — because far from the origin the leading term $a_n x^n$ swamps every other term.

So for a positive-leading-coefficient cubic ($n=3$, $a_n>0$) the graph rises from bottom-left to top-right, and because the degree is odd it is guaranteed at least one real root. That guarantee is what makes the factor theorem so useful: on a cubic there is always a root to find.

Suppose you must divide a polynomial such as $P(x)=2x^3 - 5x^2 + x - 7$ by a linear expression like $(x-3)$. Long division gets the answer but is slow and error-prone. If you only need the remainder, the remainder theorem hands it to you in one substitution.

When a polynomial $P(x)$ is divided by $(x-a)$, the remainder is $R = P(a)$.

What happens when the remainder is zero? Then the division is exact and the divisor is a factor. This is the factor theorem, and it is the bridge between the algebra of factors and the geometry of roots and $x$-intercepts.

Because $x=a$ is a root exactly when $(x-a)$ is a factor, a polynomial with known roots can be built directly: if the roots are $a$, $b$, $c$ then $P(x)=k(x-a)(x-b)(x-c)$ for some non-zero constant $k$. Reading this the other way round gives useful shortcuts. Multiplying the factors out and comparing coefficients shows that the sum and product of the roots are fixed by the coefficients alone, so you can often extract them without ever solving the equation.

For $ax^2+bx+c=0$ with roots $\alpha,\beta$: $\quad \alpha+\beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}$.

For $ax^3+bx^2+cx+d=0$ with roots $\alpha,\beta,\gamma$: $\quad \alpha+\beta+\gamma = -\frac{b}{a}, \qquad \alpha\beta\gamma = -\frac{d}{a}$.

In Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and an A-mark depends on the M-mark it follows, so a correct method protects the answer. Follow-through (FT) means a wrong value early on need not cost the marks that come after, provided the later work is correct on your own figures. But that protection only exists if the method is written down. Study how each mark below is earned by a specific line.

The factor theorem turns a cubic into a product of linear factors, which is exactly the form you need to sketch its graph, solve $P(x)=0$, and read off intercepts — the graphing skills of the rest of the functions unit. The sum and product of the roots reappear whenever a question describes the roots collectively. Master the habit — set the divisor to zero, substitute, state the conclusion, equate coefficients, show every line — and higher-degree algebra becomes variations on a method you already own.