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polynomials
finite expressions with positive whole number indices
polynomials: 5, 2x+3, 8xy-x+6
not polynomials: √5, x^-2, 4/x, 5/3
dividing polynomials
the use of long division to divide polynomials by (x±k)
-ensure the polynomial is arranged in descending order of degree of power
-divide the first term of the polynomial by x, append this to the solution
-multiply the result by (x±k)
-subtract the result from the original polynomial
-repeat this until the original polynomial is reduced to 0 or an integer remainder which is appended to the solution
expanding brackets
when mutiple polynomials in brackets are multiplied together, each term must be multiplied by each term in the other bracket(s)
factor theorem
for a polynomial f(x), if f(p) = 0, then (x-p) is a factor of f(x)
factorising polynomials
you can factorise a cubic polynomial by finding p where f(p) = 0 and dividing f(x) by (x-p)
the result of this should be factorised further if possible
quadratic equation
ax^2 + bx + c = 0
a, b and c are real constants and a != 0
quadratic roots
the solutions to the quadratic equation
discriminant
b^2 - 4ac
the value indicates how many roots the quadratic equation has:
if b^2 - 4ac > 0, there are two distinct roots
if b^2 - 4ac = 0, there is one repeated real root
if b^2 - 4ac < 0, there are no real roots
factorising
-the quadratic equation is formatted into (nx ± a)(mx ± b) = 0
-each bracket is treated as if it equals zero
-the values of x equal the values of the roots
completing the square
-the quadratic equation is formatted into a(x+d)2 ± e = 0
where d = b/2a
and e = c - b2/4a (aka c - d²)
-this can be solved giving x = ±√(e/a) - b/2a (aka ±√(e/a) - d)
-gives two values of x which equal the values of the roots
additional uses of completing the square
-to find the turning point on a quadratic graph
-to prove and/or show results using the fact that a squared term is always >= 0 i.e. k(x±a)^2 ± b always gives a result >= b
quadratic formula
(-b ± (b^2 - 4ac))/2a
gives the values of the roots
functions
functions take inputs, apply mathematical operations to it and output the value
usually denoted as f(x), or g(x) if f(x) is already specified
roots of functions
the values of x which cause an output of 0
commonly determined by solving the function's equation as quadratic(it may need to be converted first)
quadratic graphs
graphs plotted from quadratic functions, the line takes the shape of a parabola
parabola shape and QE
if a is positive the parabola will be u shaped, if it is negative it will be n shaped
x-intercept and QE
the roots of the quadratic equation equal the co-ordinates at which the parabola intersects the x axis
the number of roots determines whether the parabola crosses the x axis, merely touches it or does not touch it
y-intercept and QE
the y intercept of the parabola is equal to c
turning point and QE
the turning point of the parabola can be found by completing the square to a(x+d)2 ± e = 0 where the co-ordinates equal (-d, e)
sketching polynomials
sketching polynomials requires taking into account the following and connecting them through a smooth curve:
-y intercept
-roots
-turning points (exact values don't need to be known)
-shape
finding the y intercept of polynomials
set x to 0 where y = f(x)
finding the roots of polynomials
find values which give a y of 0 where y = f(x)
the maximum number of roots is equal to the highest power in the polynomial
odd-degree polynomials must have at least one root
turning points in polynomials
the number of turning points is equal to the highest power in the polynomial minus one
shapes of polynomials
where f(x) = ax^n + bx^n-1 ...
a positive value of a means the left end of the graph goes up for even degree polynomials and down for odd degree polynomials while a negative value means the opposite
for even-degree polynomials the right end goes in the same direction as the left end, while for odd-degree polynomials it goes in the opposite
reciprocal functions
fractions with an x term on the denominator
e.x. 1/x, 3/x^2, -2/x
reciprocal graphs
graphs of reciprocal functions- contain two asymptotes in different quadrants which never touch the x or y axis but get infinitely close at the ends
shapes of reciprocal graphs
for a/x the asymptotes are in diagonally opposite quadrants
where a is positive the asymptotes are in the bottom left and top right quadrants and where it is negative they are in the other two
for a/x^2 the asymptotes are in horizontally opposite quadrants
where a is positive the asymptotes are above y=0 and where it is negative they are below
shapes of asymptotes
the further a is from zero, the further the asymptotes are from the origin, and the closer it is the more L-shaped they are
proportional relationships
connections between two variables, can be direct or inverse
direct proportionality
y = kx
as one increases or decreases by a factor of k, so does the other
if plotted graphically, k determines the gradient of the line
inverse proportionality
y = k/x
as one increases the other decreases and vice versa
if plotted graphically, k determines the closeness of the asymptote to zero
translating graphs
f(x) + a causes the graph to move vertically by a
f(x + a) causes the graph to move horizontally by -a
stretching graphs
af(x) causes the graph to stretch vertically by a
f(ax) causes the graph to stretch horizontally by 1/a
reflecting graphs
-f(x) reflects the graph in the x axis
f(-x) reflects the graph in the y axis
linear simultaneous equations
simultaneous equations with two unknowns, both to the power of one, with one pair of solutions
can be solved by elimination or substitution
quadratic simultaneous equations
simultaneous equations where one is linear and one is quadratic, with up to two pairs of solutions
solving simultaneous equations by elimination
-multiply one or both of the equations so that the multiple of one of the unknowns matches
-subtract one equation from another to eliminate the equal unknown
-calculate the value of the second unknown using the result
-calculate the value of the first unknown by subtracting the second unknown from one of the equations
-check the results by substituting into the original equations
solving simultaneous equations by substitution
-rearrange one of the equations to make one of the unknowns the subject
-substitute this equation into the other equation
-solve the second equation to find the value of one of the unknowns
-substitute this unknown into the first equation and use that to figure out the value of the other unknown
-check the results by substituting into the oriignal equations
solving quadratic simultaneous equations
-rearrange the linear equation to make one of the unknowns the subject
-substitute this equation into the quadratic equation
-solve the quadratic equation to find the value(s) of one of the unknowns
-substitute the known unknowns into the rearranged linear equation to find the value of the other unknown
-check the results by substituting into the original equations
simultaneous equations on graphs
simultaneous equations can be plotted on graphs- linear equations will be straight lines while quadratic equations will be parabolas
the solution(s) are equal to the point(s) of intersection
linear inequalities
similar to linear equations, but with inequality signs instead therefore the solutions take ranges of values
quadratic inequalities
similar to quadratic equations, but with inequality signs instead therefore the solutions take ranges of values
solving linear inequalities
linear inequalities are solved the same way as other linear equations, but the solutions take a range using inequality signs
solving quadratic inequalities
quadratic inequalities are solved the same as other quadratic equations, but the solutions take a range
the range of solutions either take the area between the x-intercepts or outside of them, or above or below the curve
which of these ranges it takes depends on the value of a and the inequality sign, it can be determined by sketching the quadratic equation as a graph
representing solutions to inequalities
inequalities can be:
-drawn on number lines
-written normally ie 5 < x < 7
-written using set notation ie {x: x < 5} ∪ {x: x > 10}
number lines
can be used to represent solutions to inequalities
filled in dots represent <= and >= while empty dots represent < and >
set notation
can be used to represent solutions to inequalities e.x. {x: x < 3}
useful for non-graphically representing values outside of a range rather than in it ie
{x: x < 5} ∪ {x: x > 10} (x is smaller than 5 or bigger than 10, not between them)
inequalities on graphs
inequalities can be plotted on graphs
solid lines represent < and > while dotted lines represent <= and >=
the region of the graph which satisfies the inequalities is shaded
this region is determined using the fact that the solution for each line is on one side of the line for straight lines, or within a range inside or outside of the curve for parabolas
a^m x a^n
a^(m+n)
a^m / a^n
a^(m-n)
(a^m)^n
a^(mn)
(ab)^n
(a^n)(b^n)
a^-m
1/(a^m)
a^(m/n)
n√(a^m)
surds
irrational roots of integers and their multiples
√ab
√a x √b
√(a/b)
√a / √b
simplifying surds
split surds of non-prime numbers into their prime factors and multiply together any equivalent surds to produce whole numbers, the result is the sum of these
rationalising n/√a
multiply by √a
rationalising n/(√a + √b)
multiply by √a - √b
also applies if the plus and minus signs are reversed
proof by contradiction
not in the test, add later
algebraic division with no improper fractions
where F(x) and G(x) are polynomials:
F(x)/G(x) = Q(x) + r/G(x)
where Q(x) is the quotient and r is the remainder (of F(x)/G(x))
partial fractions
a fraction that has more than one linear(ax + b) factor in the denominator, that can therefore be split up into separate fractions
finding and converting split fractions
-factorise the polynomial in the denominator
-split it into the sum of proper fractions with linear denominators and algebraic placeholders for the numerators
-equate the original numerator to the sum of the algebraic placeholders multiplied by all the denominators of all other partial fractions
-choose a simple value of x to make finding the values of the algebraic placeholders easier
-done
repeated linear factors
if a squared expression appears in the factors, the unsquared expression and the squared expression will both be denominators in the split fractions as they are both factors