Polynomials and Quadratics

Question 1

When a question says solve, what does it mean?

Answer 1

Find the values of x

Question 2

How do you find the roots?

Answer 2

Set = 0
Factorise
Solve for x

Question 3

How do you find roots if the equation can’t be factorised?

Answer 3

Set = 0
Sub into quadratic formula
Solve for x

Question 4

What is the quadratic formula?

Answer 4

(-b +- v//b^2 - 4ac)/2a

Question 5

What does (x+__)^2 mean?

Answer 5

It is a repeated root

Question 6

What is the discriminant?

Answer 6

b^2 -4ac

Question 7

If the discriminant > 0, what are the roots?

Answer 7

Real and distinct roots

Question 8

If the discriminant = 0, what are the roots?

Answer 8

Real and equal roots

Question 9

If the discriminant < 0, what are the roots?

Answer 9

No real roots

Question 10

How do you determine the nature of the roots?

Answer 10

Make sure equation is in the form ax+bx+c=0
Sub equation into the discriminant
b^2 - 4ac <=> 0
statement

Question 11

How do you complete the square?

Answer 11

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)

Question 12

What are the steps in sketching a parabola?

Answer 12

'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

Question 13

How do you extract the TP from (x+p)^2 + q

Answer 13

TP (-p,q)

Question 14

How do you find the equation of parabola’s?

Answer 14

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

Question 15

What is aimed to be achieved by a quadratic inequality?

Answer 15

Find the values that x can be

Question 16

How do you solve quadratic inequalities?

Answer 16

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

Question 17

How are quadratic inequality questions asked?

Answer 17

Solve x^2 + 4x - 2 > 0

Find the values for which ………. is >0/increasing

Question 18

What are the three types of intersection?

Answer 18

b2-4ac > 0 = two intersections
= 0 = one intersection, tangent
< 0 = doesn’t intersect

Question 19

How do you find the equation of a tangent if only given the equation of another line and a gradient?

Answer 19

Sub gradient into y=mx+c
Let y=y
Sub into discriminant
Find c value
Sub c into y=mx+c

Question 20

How do you prove a line is a tangent to a parabola?

Answer 20

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’

Question 21

How do you show that something is a factor/root of an equation?

Answer 21

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'

Question 22

Hence fully factorise…..

Answer 22

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

Question 23

How would you factorise a polynomial if a factor is not given?

Answer 23

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

Question 24

What do you do if the remainder in synthetic division is not 0?

Answer 24

Add remainder outside the brackets in equation

Question 25

How do you find the value of p when given a factor?

Answer 25

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*

Question 26

How do you find the points of intersection between a line and a parabola?

Answer 26

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

Question 27

How do you find the equation of a curve?

Answer 27

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

Question 28

How do you find approximate roots?

Answer 28

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’