Polynomials and Quadratics Flashcards
When a question says solve, what does it mean?
Find the values of x
How do you find the roots?
Set = 0
Factorise
Solve for x
How do you find roots if the equation can’t be factorised?
Set = 0
Sub into quadratic formula
Solve for x
What is the quadratic formula?
(-b +- v//b^2 - 4ac)/2a
What does (x+__)^2 mean?
It is a repeated root
What is the discriminant?
b^2 -4ac
If the discriminant > 0, what are the roots?
Real and distinct roots
If the discriminant = 0, what are the roots?
Real and equal roots
If the discriminant < 0, what are the roots?
No real roots
How do you determine the nature of the roots?
Make sure equation is in the form ax+bx+c=0
Sub equation into the discriminant
b^2 - 4ac <=> 0
statement
How do you complete the square?
Open brackets, put in ax^2 and bx, leave a space, close brackets, put c outside
Divide bx by 2 to get (1), square (1) to get (2)
Put (2) in the gap and take it away outside
Put into form (x+p)^2 +q
P = (1)
Q = c+(2)
What are the steps in sketching a parabola?
'Cuts y-axis when x=0' Find the nature of the roots through b^2 - 4ac 'Cuts x-axis when y=0' - factorise Find TP by completing the square Draw and label the graph
How do you extract the TP from (x+p)^2 + q
TP (-p,q)
How do you find the equation of parabola’s?
Extract roots (x values) from given co ordinates Roots : x = \_\_\_, x = \_\_\_\_ ((If x=1 then x-1=0)) y=k(x+\_\_)(x+\_\_)(x+\_\_) Sub the y-intersect into equation (e.g. (0,12)) 12=k(\_\_)(\_\_)(\_\_) Find k Sub k into the equation Break brackets Simplify
What is aimed to be achieved by a quadratic inequality?
Find the values that x can be
How do you solve quadratic inequalities?
Will say the nature of the roots - Choose the correct sign <=> Factorise Find x values Draw an x axis and a graph Annotate the graph, >0 above graph, =0 on x-axis, <0 below graph Plot x values along x-axis If x^2 value is positive - min TP If x^2 value is negative - max TP Draw parabola Referring to the sign used in the question (<=>), decided how many lines(zones) there are in that area of the graph If 1 zone = small no < x < big no If two zones = big no < x < small no
How are quadratic inequality questions asked?
Solve x^2 + 4x - 2 > 0
Find the values for which ………. is >0/increasing
What are the three types of intersection?
b2-4ac > 0 = two intersections
= 0 = one intersection, tangent
< 0 = doesn’t intersect
How do you find the equation of a tangent if only given the equation of another line and a gradient?
Sub gradient into y=mx+c Let y=y Sub into discriminant Find c value Sub c into y=mx+c
How do you prove a line is a tangent to a parabola?
Let y=y
Solve for x
Sub x into OG equation to find y partner
‘Since there is only one point of contact at (x,y) the line is a tangent to the parabola’
How do you show that something is a factor/root of an equation?
Draw synthetic division table Put in values from equation Use factor given at the side (if x-1=0, factor =1) Add down the way, multiply diagonally 'since remainder =0, (x-\_\_) is a root'
Hence fully factorise…..
Use given factor.
Use terms at the bottom of the table to form quadratic
Factorise if possible
(Can be asked to find roots from this)
can use table function on calculator to check roots
How would you factorise a polynomial if a factor is not given?
Look at last term in equation and write out its factors
Use table function on calculator to choose a factor that =0
Complete as usual
What do you do if the remainder in synthetic division is not 0?
Add remainder outside the brackets in equation
How do you find the value of p when given a factor?
Use synthetic division Carry p term through table Remainder =0 Put entire last column into equation = 0 Solve for p Sub p into equation *Can be asked to factorise fully*
How do you find the points of intersection between a line and a parabola?
Let y=y
Set =0
Factorise - can used synthetic division by using factors of the last value in equation
Find the roots
(Sub into one of the original equations to find y-co ordinates)
Write the points of intersection
How do you find the equation of a curve?
Write out roots (x=_) Rearrange to get factors (x+_=0) y=k(x+_)(x+_)(x+_) Sub in co ordinates of a known point in Solve for k Replace k with value Simplify equation
How do you find approximate roots?
Look at scale on graph
Sub min and max value into equation = write answer
Use table function to look a 0.1 differences between min and max values
Write down the two values where the change between positive and negative takes place (e.g. f(1.2)=0.567, f(1.3)=-0.654)
Do this step until you reach the number of decimal places asked for in the question (e.g. two dp = f(1.24)
‘root exists when x=___ _dp’
