For which values of a and b , are the zeroes of q(x)=x3+ 2x2+a also the zeroes of the polynomial p (x)= x5 -x4-4x3+3x2+3x+b? Which zeroes of p (x) are not the zeroes of q (x)?
To determine the values of a and b for which the zeroes of q(x) are also zeroes of p(x), we need to consider the common factors of the two polynomials.
First, let's find the zeroes of q(x) by setting it equal to zero: q(x) = 0.
This gives us the polynomial equation x^3 + 2x^2 + a = 0.
To find the zeroes of this cubic polynomial, we can use various methods like factoring or synthetic division, but for this explanation, we'll use the rational root theorem.
According to the rational root theorem, the rational zeroes of a polynomial equation are all possible ratios, where the numerator is a factor of the constant term (a) and the denominator is a factor of the leading coefficient (1 in this case).
So, the possible rational zeroes of q(x) = x^3 + 2x^2 + a are of the form p/q, where p is a factor of 'a' and q is a factor of '1'.
Now, let's consider p(x) = x^5 - x^4 - 4x^3 + 3x^2 + 3x + b.
For the zeroes of p(x) to be the zeroes of q(x), they must be the common roots.
Therefore, the common zeros of p(x) and q(x) are the rational roots of q(x) that also satisfy the equation p(x) = 0.
To determine the zeros of p(x), we can set it equal to zero: p(x) = 0.
Now, let's check which of the rational roots of q(x) satisfy p(x) = 0.
If any of the rational roots of q(x) also satisfy p(x) = 0, then those values of a and b will be the solution.
The zeroes of p(x) that are not the zeroes of q(x) will be the remaining roots of p(x) that do not satisfy q(x) = 0.
To summarize:
1. Find the rational roots of q(x) = x^3 + 2x^2 + a using the rational root theorem.
2. Substitute each potential rational root into p(x) = x^5 - x^4 - 4x^3 + 3x^2 + 3x + b to see if it satisfies the equation.
3. The values of a and b for which the zeroes of q(x) are also zeroes of p(x) will be the ones where the potential rational roots satisfy p(x) = 0.
4. The zeroes of p(x) that are not the zeroes of q(x) will be the remaining roots of p(x) that do not satisfy q(x) = 0.
If the roots of q are roots of p, then q must divide p. If you do the long division, you will find that the remainder is
-(a+1)x^2 + 3(a+1)x + (b-2a)
These must all be zero, which means that
a = -1
b = -2
So,
p(x) = x^5-x^4-4x^3+3x^2+3x-2
q(x) = x^3+2x^2-1
q(x) = (x+1)(x^2+x-1)
p(x) = (x+1)(x^2+x-1)(x-1)(x-2)