Consequences of the factor theorem

Factoring after finding several zeroes, the bound of n zeroes for a degree n polynomial, the test for identical polynomials, and multiple roots as tangents when finding intersections.

Several zeroes at once

Example 1●●●

Show that $2$ and $-1$ are zeroes of $P(x) = x^4 - x^3 - 3x^2 + x + 2$, and factor it fully.

$P(2) = 16 - 8 - 12 + 2 + 2 = 0$ and $P(-1) = 1 + 1 - 3 - 1 + 2 = 0$.

$P(x) = (x^2 - x - 2)(x^2 - 1)$

So (x − 2)(x + 1) = x² − x − 2 is a factor. Divide, or match coefficients.

$P(x) = (x - 2)(x - 1)(x + 1)^2$, with a double zero at $-1$.

How many zeroes?

Two useful consequences follow.

If a polynomial of degree at most $n$ has more than $n$ zeroes, it must be the zero polynomial.
If two polynomials of degree at most $n$ agree at $n + 1$ (or more) values of $x$, they are identically equal. For example, a quadratic is completely determined by three points.

Multiple roots and tangents

To find where two curves meet, form the equation $f(x) = g(x)$ and solve. The multiplicity of each root tells you how they meet.

Where does y = mx − 2 meet y = x³?

Orange: y = x³. Blue: the line y = mx − 2, which always passes through (0, −2). At m = 3 two crossing points merge into a tangent.

Try this

Slide $m$ slowly through $3$. Watch two intersection points come together and then disappear.

Example 2●●

Show that $y = 3x - 2$ is a tangent to $y = x^3$, and find where else they meet.

Solve $x^3 = 3x - 2$: $x^3 - 3x + 2 = 0$.

$(x - 1)(x^2 + x - 2) = (x - 1)^2(x + 2) = 0$

x = 1 is a zero. Divide by x − 1.

Double root $x = 1$: tangent at $(1, 1)$. Simple root $x = -2$: the line crosses the curve again at $(-2, -8)$.

Flashcards

Distinct zeroes $a_1, ldots, a_k$

$(x - a_1)\cdots(x - a_k)$ is a factor.

Zeroes of a degree $n$ polynomial

At most $n$.

Double root of $f(x) = g(x)$

The curves are tangent there.

Multiple zero test

$P(a) = 0$ and $P'(a) = 0$.

Practice

What is the largest number of distinct zeroes a quartic can have?
$2$ and $-1$ are zeroes of $P(x)$. Which must be a factor?
Factor $x^4 - x^3 - 3x^2 + x + 2$ fully.
Two cubics agree at $5$ different values of $x$. Then they:
Which statement is true?
$y = 3x - 2$ is tangent to $y = x^3$ at $x = 1$. At what other $x$-value do they meet?
The equation for the intersections of two curves has a double root at $x = a$. At $x = a$ the curves:
$P(x) = (x - 2)(x - 1)(x + 1)^2$. What is the multiplicity of the zero $-1$?
$P(a) = 0$ and $P'(a) = 0$ means $a$ is: