Roots of Polynomials

Question 1

how many complex roots must there be to a real polynomial

Answer 1

must be an even number due to conjugate pairs

Question 2

how can a quadratic be represented with alpha and beta

Answer 2

p(x - a)(x - b)
i.e
px2 - p(a + b)x + pab

Question 3

sum of root singles

Answer 3

-b/a

Question 4

sum of products of root pairs

Answer 4

c/a

Question 5

when making an equation what should you replace a with

Answer 5

1

Question 6

what can a cubic equation be rewritten as with roots

Answer 6

p(x -

Question 7

what can a cubic equation be rewritten as with roots

Answer 7

p(x - a)(x - b)(x - y)
i.e
px3 - p(a + b + y)x2 + p(ab + by + ay)x - paby

Question 8

sum of products of triples

Answer 8

-d/a

Question 9

How to make a new equation with modified roots

Answer 9

either sub in the new roots to the sum formulae
or make a new variable w relating to the change of roots then substitute into original

Question 10

what can a quartic equation be rewritten as with roots

Answer 10

p(x-a)(x-b)(x-y)(x-s)
px4 - p(a + b + y + s)x3 + p(ab + ay + as + by + bs + ys)x2 - p(aby + abs + ays + bys)x + pabys

Question 11

sum of product of quadruples

Answer 11

e/a

Question 12

if you are given one complex root what can you assume about another root

Answer 12

it is a conjugate