Polynomials and Quadratics

Question 1

shape of graph of a quadratic

Answer 1

parabola (smile=minTP, frown=maxTP)

Question 2

quadratic formula

Answer 2

x=-b+/-square root of b squared - 4ac all over 2a

Question 3

discriminant formula

Answer 3

b squared - 4ac

Question 4

completing the square

Answer 4

in the form y=a(x+p)squared +q
axis of symmetry x=-p
turning point (-p,q)
*can check answer by expanding brackets*

Question 5

steps of sketching parabola

Answer 5

1. find discriminant
2. cuts y-axis, x=0
3. cuts x-axis, y=0
4. calculate axis of symmetry/minTP

Question 6

determining equation of parabola

Answer 6

y=k(x-a)(x-b)

1. work out roots
2. factors
3. sub point in to work out k
4. write equation

Question 7

solving quadratic inequalities

Answer 7

need to know shape (max or min) and roots
sketch graph
write out answer

Question 8

steps on how to prove tangency and find point of contact

Answer 8

1. equal equations to each other
2. equal to zero
3. factorise and find roots (should be repeated root)
4. sub x coordinate (root) into one of the equations to find y coordinate

Question 9

steps on how to find equation of tangent

Answer 9

1. sub gradient into y=mx+c
2. equal equations to each other
3. equal to zero
4. discriminant is equal to zero, find c
5. find equation using y=mx+c

Question 10

degree of a polynomial

Answer 10

value of the highest power

Question 11

synthetic definition steps

Answer 11

1. write coefficients across top (put zero if there isn’t one)
2. write number you want to evaluate at left
3. bring down first coefficient
4. multiply by side number, add onto coefficients
5. number at end is remainder

Question 12

sentence at end of synthetic division

Answer 12

since remainder is zero, x-4 is a factor and x=4 is a root

Question 13

factorising polynomials

Answer 13

use factor found in synthetic division and numbers on the bottom of synthetic division
factorise or repeat synthetic division if necessary
use number at end of polynomial to find factors

Question 14

using synthetic division to solve equations

Answer 14

exact same steps as before but find roots of the factors at the end (equal them to zero)

Question 15

quotient and remainder

Answer 15

quotient are the numbers at bottom of synthetic division and remainder is number at end

Question 16

finding unknown coefficients

Answer 16

1. use synthetic division
2. equal remainder to zero
3. rearrange to find unknown

Question 17

finding multiple unknown coefficients

Answer 17

exactly the same as before but use simultaneous equations at the end.

Question 18

determining equation of a curve (when roots are known and at least one other point)

Answer 18

1. right out roots
2. rearrange to get factors
3. write out y=k(x-a)(x-b) with factors
4. sub in known point to find k
5. write out final equation
   * if repeated root, write twice in equation*

Question 19

what is a polynomial?

Answer 19

an expression containing the sum or difference of algebraic terms with powers or the equivalent in factorised form.

Question 20

when trying a divisor of the form (x-a) it is usually a good idea to start with…

Answer 20

(x-1)

Question 21

how to determine the equation of a polynomial from its graph

Answer 21

1. use the roots to determine factors eg roots a,b,c give factors (x-a)(x-b)(x-c)
2. remember the polynomial may have a scalar, k eg y=k(x-a)(x-b)(x-c)
3. substitute coordinates of y-intercept to determine k.

Question 22

how to determine points of intersection of line and polynomials

Answer 22

1. make the equations equal and rearrange to equal 0.
2. factorise using synthetic division
3. identify and interpret roots –> equal roots are required for tangency
4. find any other points of intersection

Question 23

completing the square is used to:

Answer 23

1. determine the coordinates of the turning point
2. solve a quadratic equation which:
   - does not factorise
   - is undefined using the quadratic formula
   - has discriminant<0, no real roots
   - gives a parabola that lies entirely above or below x-axis.

Question 24

what form does a quadratic need to be written in to complete the square?

Answer 24

ax^2+bx+C

Question 25

quadratic equations can be solved by:

Answer 25

1. using the graph
2. factorising
3. using quadratic formula
4. completing the square

Question 26

how to solve quadratic inequalities

Answer 26

1. solve the related quadratic equation to find the roots
2. make a sketch of the graph of the quadratic function
3. identify the solution of the inequality:
   f(x)<0 has solutions below x-axis
   f(x)>0 has solutions above x-axis

Question 27

if b squared -4ac<0…

Answer 27

no real roots

Question 28

if b squared -4ac=0…

Answer 28

real and equal roots

Question 29

if b squared -4ac>0

Answer 29

real and distinct roots

Question 30

how to determine relationship between a line and a parabola

Answer 30

1. make equations equal and rearrange to 0
2. find and interpret the discriminant of the resulting quadratic equation
   =0 tangent
   >0 crosses parabola in 2 places
   <0 does not touch parabola
3. find the coordinates of any points of contact