3. Find Q(7) if Q(x) = x^7 - 48x^5 - 37x^4 + 210x^3 + 11x^2 - 80x + 23 by hand. Show your work. You may use any theorem. (Rational Zeros Theorem, Factor Theorem, Remainder Theorem).
4. What are the possible rational zeroes of Q(x) from the previous problem?
Please explain in great detail!
The problem
says to use the "rational zeros theorem, factor theorem, remainder theorem". Do you not know what those are? Haven't you been attending class?
The rational zeros theorem says that any rational zero of a polynomial, with integer coefficients, must be of the form x= p/q where the denominator, q, evenly divides the leading coefficient and the numerator, p, divides the constant term. Here, the leading coefficient is 1 so the denominator must be either 1 or -1: any rational zero must be an integer. The constant term is 23 so any rational zero must be 1, -1, 23, or -23. Of course, the rational zero theorem
doesn't say that the
is a rational zero- so check those four numbers to see if one is a zero.
The "factor theorem" says that if x= a is a zero of a polynomial, then x- a is a factor of the polynomial. If you have been able to find a rational root, say x= p/q, then x- p/q is a factor which is equal to (1/q)(qx- p) so that qx- p is a factor. Divide the polynomial by that to reduce the degree of the polynomial so you can look for other zeros.
The "remainder theorem" says that if you divide a polynomial by x- a, the remainder will be the polynomial
evaluated at x= a. That leads to the "factor theorem" above.
