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 ax^j 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 see that (x-2)³ = x³ - 6x² + 12x - 8 is in fact a polynomial.

Note:

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) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

where each of the a's are fixed real numbers. If any of the a's are 0 then the terms are omitted. When a_n is not 0, then we say that the polynomial p(x) has DEGREE n. When evaluating the polynomials at a certain value of x all we do is substitute in the number any where we see an x. Here are some examples of what we mean. The real number that satisfies the equation p(x) = 0 is known as the root of the equation.Example.

Note:

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 p(x) = ax² + bx + c where a 0 then

(i)

if b² - 4ac < 0 then p(x) has no real roots.

(ii)

b² - 4ac = 0 then -b/2a is the only root.

(iii)

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 x =

What if we are given r is a root of a polynomial p(x)? Then we can say that (x-r) is a factor of p(x) and p(x) = (x-r)q(x), where q(x) is another polynomial of degree 1 less than p(x). Examples.

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