POLYNOMIALS
A polynomial, of degree n, is a function of the form:
where
are constants
We can use synthetic division to divide a polynomial by a linear function.
e.g. 3x3 – 5x + 7 ÷ (x + 2)
-2
3
3
0
-6
-6
Remember to include zero coefficients
-5
7
12 -14
7
-7
Remainder
These are the coefficients of the quadratic
So 3x3 – 5x + 7 ÷ (x + 2) = 3x2 – 6x + 7 remainder (-7)
or 3x3 – 5x + 7 = (3x2 – 6x + 7)(x + 2) – 7
The Remainder Theorem states that if a polynomial f (x) is divided by (x – h) the
remainder is f (h).
The Factor Theorem states that if f (h) = 0 then (x – h) is a factor of f (x), and
conversely, if (x – h) is a factor of f (x) then f (h) = 0.
Factorising Polynomials
We can use synthetic division to factorise a polynomial function.
e.g. Factorise 2x3 + x2 – 50x – 25
Look for factors that could multiply to give the constant term of -25. i.e. try (x + 1),
(x – 1), (x + 5), (x – 5), (x + 25), (x – 25)
Try (x + 1) first:
-1
2
2
1
-2
-1
-50 -25
1 49
-49 24
Remainder
0 so not a factor
Try (x + 5) next:
-5
2
2
1
-10
-9
-50 -25
45 25
-5
0
Remainder = 0 so (x + 5) is a factor
Now factorise the quadratic
So 2x3 + x2 – 50x – 25 = (2x 2 – 9x – 5)(x + 5)
= (2x + 1)(x – 5)(x + 5)
Solving Polynomial Equations
We can use synthetic division to solve polynomial equations.
e.g. Solve 2x3 + x2 – 50x – 25 = 0
We can use synthetic division to factorise the polynomial (see example above).
2x3 + x2 – 50x – 25 = 0
(2x + 1)(x – 5)(x + 5) = 0
2x + 1 = 0
or
x–5=0
or
x+5=0
x=-½
x=5
x = -5
Approximate Roots
We can approximate the roots of a polynomial using an iterative method.
e.g. Show that the curve y = x3 – 4x2 – 2x + 7 has a root between 1 and 2. Find this
root to 1 decimal place.
when x = 1, y = 13 – 4×12 – 2×1 + 7 = 2 this is a point above the x-axis
when x = 2, y = 23 – 4×22 – 2×2 + 7 = -5 this is a point below the x-axis, hence the
graph has a root between 1 and 2.
when x = 1.2, y = 1.23 – 4×1.22 – 2×1.2 + 7 = 0.568 this is a point above the x-axis
when x = 1.3, y = 1.33 – 4×1.32 – 2×1.3 + 7 = -1.63 this is a point below the x-axis,
hence the graph has a root between 1.2 and 1.3.
when x = 1.25, y = 1.253 – 4×1.252 – 2×1.25 + 7 = 0.203 this is a point above the xaxis, hence the graph has a root between 1.25 and 1.3.
The root is 1.3 (1 decimal place)
Graphs of Polynomials
y=
A horizontal line through the point (0,
y=
).
(y = mx + c)
A linear function which is increasing when
and decreasing when
.
y=
A quadratic function which has a minimum turning point when
point when
.
y=
A cubic function.
and a maximum turning
Polynomials Practice
http://www.bbc.co.uk/bitesize/higher/maths/algebra/polynomials/revision/1/
http://www.bbc.co.uk/bitesize/higher/maths/algebra/polynomials2/revision/1/
Revise Polynomials and try the TESTBITE.