Polynomials and Roots
We first must note that when we use a letter like x or t in an expression, it often denotes a variable real number. If the expression contains only addition and multiplication as the types of arithmetic operations used, then the expression is called a POLYNOMIAL. Examples.
Each term in a polynomial can be written as axj where a is a real number and j is a non-negative integer (ie. j in N°). When there is just a single number as the 2 in example 1 above, we refer to this as the constant term of the polynomial and is thought of as 2x0.
By expanding out expressions of the type (x-2)3, for example, we
- Usually, polynomials are written in standard form with the powers of the variable in decreasing order.
Polynomials are named like p(x), q(t), h(z), etc. So p(x) is a paticular polynomial with the variable being x. For instance, we could write:
- p(x) = anxn + an-1xn-1 + ... +
a1x + a0
- If p(x) = ax+b with a not 0, then p(x) has exactly one root, which is -b/a
When we have polynomials of degree 2, we can find the roots by COMPLETING THE
SQUARE. This is best explained with an
example. Sometimes, for the degree 2
polynomial, there is only one root. In fact, there might not be any at all!
Let's look here at some examples. There is
an easy trick to see how many roots a degree 2 polynomial has. If
- if b2 - 4ac < 0 then p(x) has no real roots.
- b2 - 4ac = 0 then -b/2a is the only root.
- if b2 - 4ac > 0 then are the two roots of p(x). Proof.
These results are combined into what is called the QUADRATIC FORMULA:
If ax2 + bx + x = 0 then