Quadratics and cubics are particular examples of polynomial functions.
a quadratic function is a polynomial of degree 2
a cubic has degree 3.
In general a polynomial of degree n has the form
a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0
where a_0, a_1, ... a_n are real numbers, and a_n ≠ 0 .
When solving cubic equations, the spotting of a root leads immediately to a simple linear factor. For example,
3 gives a zero value for the cubic x^3 - 5x^2 + 11x - 15
⇒ (x - 3) is a factor of x^3 - 5x^2 + 11x - 15
⇒ x^3 + 5x^2 + 11x - 15 = (x - 3) × (\text{quadratic})
The same technique can be applied to polynomials of any degree. For example,
2 gives a zero value for the function x^4 + 3x^2 - 17x + 6
⇒ (x - 2) is a factor of x^4 + 3x^2 - 17x + 6
⇒ x^4 + 3x^2 - 17x + 6 = (x - 2) × (\text{cubic})
In general
If P(x) is a polynomial of degree n and has a zero at x = a,
that is P(a) = 0, then (x - a) is a factor of P(x) and
P(x) = (x - a) × Q(x)
where Q(x) is a polynomial of degree (n - 1).
This is known as the factor theorem.
Worked Examples
Show that (x - 5) is a factor of the polynomial
x^5 - 4x^4 - x^3 - 21x^2 + 25
| P(5) | = 5^5 - 4 × 5^4 - 5^3 - 21 × 5^2 + 25 |
| = 3125 - 2500 - 125 - 525 + 25 | |
| = 0 |
Show that x = \frac{1}{2} gives a zero value to the 2x^3 - 3x^2 - 3x + 2.
Also show that 2x^3 - 3x^2 - 3x + 2 has two linear factors other than x = \frac{1}{2}.
We write the function as P(x) = 2x^3 - 3x^2 - 3x + 2
We can see that
| P\left(\frac{1}{2}\right) | = 2\times\left(\frac{1}{2}\right)^3-3\times\left(\frac{1}{2}\right)^2-3\times\left(\frac{1}{2}\right)+2 |
| = \frac{1}{4}-\frac{3}{4}-\frac{3}{2}+2 | |
| = 0 |
We know that \left(x - \frac{1}{2}\right) is a factor so (2x - 1) is also a factor and we can write
2x^3 - 3x^2 - 3x + 2 = (2x - 1) × (ax^2 + bx + c)
Here a, b and c are constants and so a = 1 and c = -2. Comparing coefficients of x on each side of the identity, -3 = -b + 2c, giving b = -1.
You can check this by comparing coefficients of x^2.
So
| P(x) | = (2x - 1) × (x^2 - x - 2) |
| = (2x - 1) × (x - 2) × (x + 1), after factorising the quadratic. |
Exercises
