Question

What polynomial has roots of -6 -4 and 1?
a. x^3-9x^2-22x+24
b. x^3-x^2-26x-24
c.x^3+x^2-26x+24
d.x^3+9x^2+14x-24

Answer 1

the polynomial shall be (x+6)(x+4)(x-1) . On solving it , you will get

Answer 2

@Aadi4502 how did you get that answer?

Answer 3

Well, for any equation to have its roots as a, b,c. The equation must give a value of 0 when x is put to be a or b or c. So here, we have to find an equation with roots -6, -4, and 1 meaning, if I put the value of x=-6, then I should get y=0. same goes for -4 and 1. So, the equation would be (x+6)(x+4)(x-1).

Answer 4

@Aadi4502 okay but how did you get b as your answer?

Answer 5

factor it and set it equal to 0

Answer 6

@Aadi4502 Im sorry im just really confused math isn't my greatest subject

Answer 7

Ok i made a calculation mistake, the answer is d. Shall i explain the multiplication ?

Answer 8

@Aadi4502 yes please

Answer 9

Well then, lets first write the proposed equation ie (x+6)(x+4)(x-1). now in order to multiply it, (going by the basic versions), you open up the first two brackets ie (x+6)(x+4) which would give you x^2+6x+4x+6*4 = x^2+10x+24. Now you multiply this value with the third bracket ie (x^2+10x+24)*(x-1). You will get x^3+10x^2+24x -x^2-10x-24 = x^3+9x^2+14x-24. Hence your answer d. Tell me if you have more doubts regarding the problem

Answer 10

d.\[(x+6) (x+4) (x-1)=x^3+9 x^2+14 x-24 \]

Answer 11

There are a couple of ways.
One is factor theorem:

If (x - a) is a factor of the polynomial, then P(a) = 0.
Conversely if x=a is a root of the polynomial i.e. P(a) = 0, then (x-a) is a factor of P(x).

This means if you sub a number into your polynomial and it gives zero, it is a root.
The other way is to factorise your polynomial and find you roots from there using the same theorem.

The third method is to use relationships between roots which focuses on the coefficients of your polynomial.
For a cubic written as ax^3 + b^2 + cx + d, the sum of roots is:
\[\alpha + \beta + \gamma = \frac{ -b }{ a }\]
The product of roots at a time is
\[\alpha \beta \gamma = \frac{ -d }{ a }\]
The sum of roots two at a time is:
\[\Sigma (\alpha \beta) = \alpha \beta + \alpha \gamma + \beta \gamma = \frac{ c }{ a }\]
Then compare your roots eg. -6, -4, 1 sum one at a time to (-6-4+1)= -9 and give product of 24 and sum two at a time (-6 x -4) + (-6 x 1) +(-4 x 1) = 24 - 6 - 4 = 14.
This corresponds to answer d if you look at the sum and product of roots formulae.