try to factorise it first, if it is not already factorised.Writing a polynomial as a product of factors --for example - makes it easier to identify the roots – the pointswhereTheroots will be points on the x – axis, since each root is asolution of the equation
Find the–intercept by substitutinginto
Decide whether the curve tends toorasx tends toorIfthe coefficient of the highest power ofispositive when the expression is expanded, then as tendstosodoesandif the coefficient is negative, then astendsto tendsto
Each distinct root --with no power - will give rise to a point on the–axis where the curve CROSSES the–axis, and each repeated root - given by a factor– will give rise to a point on the curve which touches the–axis but does not cross it ifiseven, or which forms a tangent to the–axis and crosses it ifisodd. Examples are shown below.
For example, to sketch
The roots are the solutions to
These are
Substitutingintothe expression gives
The highest power ofisandthe coefficient ofis2 (considersoas
Each root is distinct, so the graph crosses the–axis at each root. The graph is sketch below.
To sketch
The roots are given by
Substitutingintothe expression gives
The highest power ofisandthe coefficient ofis-1 (considersoas
The root atisa double root, so the curve touches theaxisatbutdoes not cross it, and the root atisa single root so the graph crosses the–axis there.
The curve is sketched below.
]]>Example: Show thatis a factor of
henceis a factor of
Example:andare both roots of the quadratic expressionFind
Sinceandare both rootsandare both factors. A quadratic equation has at most two real factors/roots hence
More complicated questions may involve simultaneous equations:
andare both roots ofFind a and b and factorise f(x)..
divided byremainder is
divided byremainder is
We now solve the simultaneous equations
(1)
(2)
3*(1)+(2) gives
Then from (1)
is a cubic and has the two given roots so must have a third linear factor
by considering the coefficient ofand by considering the constant term 1
]]>Example:Ifis a root offactorise
Ifis a root,is a factor.
constant:
hence
The quadratic in the brackets can now be factorised by inspection:
Example:Ifis a root offactorise
Ifis a root,is a factor.
constant:
hence
The quadratic in the brackets can now be factorised by inspection:
]]>
Suppose thatandTo find the coefficient ofin the binomial expansion ofputthen the coefficient ofis and the coefficient ofis
If bothandare terms in x then things get slightly more complicated.
Suppose we want to find the coefficient ofin the binomial expansion of
andso the coefficient ofisand the (r+1) ^{th }term is
Putthenso the coeeficient ofis
]]>Since the coefficients of the polynomial are real, the purely complex rootsand occur as a complex conjugate pair so that we can writeandIf the real root isthe the polynomial can be written as
Then from considering the coefficients of
(1)
(2)
and the constant term gives(3)
(3) divided by (2) gives
Then from (1)
Then from (2),
Then
]]>This means we need five conditions to determine the equation of the quartic. In fact we always need five condtions, but some of these may be hidden. If the equation of the quartic includes a repeated factorthen the corresponding condition would be ' the graph of the quartic is a tangent at'. This statement includes two conditions.
1. is a tangent
2. at
Taking this sort of thing into account, we can find the equation of the quartic.
Example: Find the quartic which touches the x - axis at 3, cuts it at -2 and also passes
throughand
1. ' touches the x - axis at 3' implies the quartic is a tangent atso includes a factor
2. ' cuts it at -2' implies a factor
The quartic must then take the form
passes throughso (1)
passes throughso (2)
(1) divided by (2) gives
Then from (2)
The equation of the quartic is
]]>At each stage we work to eliminate the highest power ofTo start with the highest power ofis 4: multiply the denominatorby– the first term of the quotient - to getand subtract from the numerator to getNow the highest power ofis 3. We multiply the denominator by– the nest term of the quotient - to getand subtract to getThe highest power ofis 2. We multiply the denominator by 2 – the last term in the quotient - to getand subtract to get zero. HenceThere is no remainder.
If instead we are finding the quotient and remainder ofwe follow the same process, but now, as shown below, the remainder is 9 hence the division is now
hence
]]>The diagram below shows a ladder leaning against a wall. The ladder is 10m long and rests against a 1 m ^{3 } box. The ladder reaches x m up the wall. Find
We can extract similar triangles from the above diagram and use the ratios of their lengths to derive a polynomial equation in
From the diagram above, using Pythagoras theorem to findgives so base divided by height gives
From the diagram above base divided by height givesThese two ratios must be the same since the triangles are similar so
Rearrangement of this expression gives
The graph ofis shown below. The required root is obviously atThe root can be found using a graphical calculator.
]]>Proof
Letbe a polynomial and letbe a number.
Ifdividesthen the remainder on division ofbyis zero and there is a polynomialsuch thatso thatandis a root of
Now assume thatis a root ofso thatPerform long division ofbyto obtain quotientand remainderthen write
(1)
The degree ofis less than the degree ofsoso is just a constant. WriteNow substituteinto (1) to give
so thatand we can write
]]>