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Question 477325: Can you please help me solve those problems!! Thank you very much!!!!
1)Let f(x) = x^4 -3x^2 + 2 and g(x) = 2x^4 -6x^2 + 2x -1.
(a) What is the degree of f(x) * g(x)?
(b) Let a be a constant. What is the largest possible degree of f(x) + a * g(x)?
(c) Let b be a constant. What is the smallest possible degree of f(x) + b * g(x)?
2)Find t if the product of x^3 - 4x^2 + 2x - 5 and x^2 + tx - 7 has no x^2 term.
3)Assume that f(3) = 4. Find a point on the graphs of each of the following:
(a) y = f(x) + 4
(b) y = f(x + 1)
(c) y = -2f(2x - 2)
(d) y = (1/2)f( x/2 )
4)Multiplication is commutative because x*y = y *x for all x and y. Multiplication is associative
because x *(y *z) = (x *y) *z. The identity of multiplication is 1, because x*1 = x for all x.
Let a & b = a + ab + b for all a and b.
(a) Is the operation & commutative?
(b) Is the operation & associative?
(c) What number is the identity of &?
(d) What number is the inverse of 3 with respect to &? (Recall that the inverse of 3 with
respect to multiplication is 1/3 because 3 * 1/3 equals the identity of multiplication.)
Thank you very much for your help!!!! I appreciate it a lot!!!!!!! Thanks!!!
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Please try not to write too many problems into one post; tutors are less likely to answer all of them. I'll get you started on them, but you can finish the rest.
1. (x^4 - 3x^2 + 2)(2x^4 - 6x^2 + 2x - 1) = 2x^8 + stuff. What is the degree? For parts b) and c) note that multiplying a polynomial by a constant does not change the degree. But you are adding a*g(x) to some other polynomial of the same degree, in which there exists a constant a such that the highest exponent terms cancel...
2. This one, you will have to expand, collect all the x^2 terms, and set the coefficient equal to 0. It should be some expression in terms of t, equals zero, solve for t.
3. You know f(3) = 4, but nothing else. How else would you locate a definitive point unless you set the argument of the function equal to 3?
4. The operation is commutative if a & b = b & a, and associative if (a & b)& c = a & (b & c). c) S is an identity element of & if a & S = a for all a, i.e. a + aS + S = a. d) z is the inverse if 3 & z = S, where S is the identity element. You will have solved for S from part c), so set 3 & z equal to S and solve.
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