if alpha and beta are zeroes of the polynomial x2-5x+6,then find the polynomial whose zeroes are root 3 and -root 3
" ..... find the polynomial whose zeroes are root 3 and -root 3"
y = (x-√3)(x+√3)
= x^2 - 3
What does the fact that alpha and beta are zeros of the the polynomial x^2 - 5x + 6 have to do with it?
other than alpha = 2 and beta = 3
by splliting middle term
Math
To find the polynomial whose zeroes are root 3 and -root 3, we can use the fact that if alpha and beta are zeroes of a quadratic polynomial ax^2 + bx + c, then the polynomial can be factored as (x - alpha)(x - beta).
In this case, the given quadratic polynomial is x^2 - 5x + 6, so the factors are (x - alpha)(x - beta). To find the values of alpha and beta, we can compare the given polynomial with the general form, ax^2 + bx + c.
Comparing x^2 - 5x + 6 with (x - alpha)(x - beta), we can equate the corresponding coefficients:
1 = alpha + beta (equation 1)
-5 = -(alpha + beta) (equation 2)
6 = alpha * beta (equation 3)
From equation 2, we get alpha + beta = 5.
Now, let's solve equations 1 and 3 simultaneously to find the values of alpha and beta.
From equation 1, we have alpha = 5 - beta.
Substituting alpha = 5 - beta in equation 3, we can solve for beta:
6 = (5 - beta) * beta
6 = 5beta - beta^2
beta^2 - 5beta + 6 = 0
This quadratic equation can be factored as (beta - 2)(beta - 3) = 0.
So, the values of beta are 2 and 3.
Substituting beta = 2 and beta = 3 in equation 1, we can find the corresponding values of alpha:
When beta = 2, alpha = 5 - 2 = 3.
When beta = 3, alpha = 5 - 3 = 2.
Therefore, the values of alpha and beta are 3 and 2, respectively.
Now that we have the values of alpha and beta, we can write the polynomial whose zeroes are root 3 and -root 3 as:
(x - 3)(x - 2) = (x^2 - 5x + 6)
So, the polynomial whose zeroes are root 3 and -root 3 is x^2 - 5x + 6.