Polynomials Expressions like 3x 4 + 2x 3 – 6x and m 6 – 4m 2 +3 are called polynomials. (5x – 2)(2x+3) is also a polynomial as it can be written 10x x - 6. The degree of the polynomial is the value of the highest power. 3x 4 + 2x 3 – 6x is a polynomial of degree 4. m 6 – 4m is a polynomial of degree 6. In the polynomial 3x 4 + 2x 3 – 6x , the coefficient of x 4 is 3 and the coefficient of x 2 is -6.

A root of a polynomial function, f(x), is a value of x for which f(x) = 0

Hence the roots are 2 and –2.

Division by (x – a) Dividing a polynomial by (x – a) allows us to factorise the polynomial. You already know how to factorise a quadratic, but how do we factorise a polynomial of degree 3 or above? We can divide polynomials using the same method as simple division Conversely, Divisor Quotient Remainder

Divisor Quotient Remainder Note: If the remainder was zero, (x – 2) would be a factor. This is a long winded but effective method. There is another.

Let us look at the coefficients quotient divisor remainder This method is called synthetic division. If the remainder is zero, then the divisor is a factor. Let us now look at the theory.

We divided this polynomial by (x – 2). 62 If we let the coefficient be Q(x), then Remainder Theorem If a polynomial f (x) is divided by (x – h) the remainder is f (h). We can use the remainder theorem to factorise polynomials.

Remainder Theorem If a polynomial f (x) is divided by (x – h) the remainder is f (h) Since the remainder is zero, (x – 4) is a factor.

To find the roots we need to consider the factors of

To find the roots we need to consider the factors of

Finding a polynomial’s coefficients We can use the factor theorem to find unknown coefficients in a polynomial. Since we know (x + 3) is a factor, the remainder must be zero p p -3p 4 - 3p 9p p - 27

2 1 a -1 b a 4 + 2a 3 + 2a 6 + 4a 6 + 4a + b a + 2b 4 + 8a + 2b = a -1 b a a a 16a a + b a - 4b a - 4b = 0

Solving polynomial equations If we sketch the curve of f (x), we see that the roots are where f (x) crosses the X axis.

Functions from Graphs f (x) x d a b c The equation of a polynomial may be established from its graph.

1. From the graph, find an expression for f (x). f (x) x Substituting (0, -12)

2. From the graph, find an expression for f (x). f (x) x Substituting (0, 30)

Curve sketching The factor theorem can be used when sketching the graphs of polynomials. The Y axis intercept is (0, 12) Y axis Intercept

We will use synthetic division to find the roots of the function. This will tell us where the graph crosses the X axis X axis Intercept

Stationary Points

Nature of Stationary Points Slope

Approximate Roots When the roots of f (x) = 0 are not rational, we can find approximate values by an iterative process. We know a root exists if f (x) changes sign between two values. f (x) x a b f (x) x a b A root exists between a and b.

Hence the graph crosses the x - axis between 1 and and and and and and and and and and and 1.28 Hence the root is 1.28 to 2 d.p.