Pls answer these....?

Question

1.What function is defined by P(x)=a (sub n) x^n + a(sub n-1) x^x-1 +.....a (sub 1) x+a?
2.Find the values of the constant A, B and C so that x(2x^3+7x-5)=Ax^4+bx^3+Cx^2-5x is an identity.
3. Given f(x)=x^4+4x^3+2x^2+x-3, find the remainder when f(x) is divided by x-1?
4. Factor 2x^3-5x^2-x+6 completely.
5. If (x+1) is a factor of a^2x^3+ax+2, find the value of a.
6. Draw the graph of y=x^3+x^2-2x by finding the x- and y- intercepts.
7. How many variations in sign are there in the polynomial x^4-5x^3+2x^2+4x-7?
8. How many roots does the equation 2x^8-7x^2-3x+1=0 have/
9. Give the complete set of possible rational roots of 2x^5-4x^3+7x+10=0.
10. What is m if x+3 is a factor of 3x^2+2x+m?

Answer 1

1. P is a polynomial. The degree of P is n if a_n different of 0

2. x(2x³ + 7x - 5)= 2x^4 + 7x² - 5x = ax^4 + bx³ + cx² - 5x
then a = 2
b = 0
c = 7

3. if x = 1, f(1) = 5 then the remainder is 5

4. 2x³ - 5x² - x + 6 = 0 if x = -1
then a factor is x + 1
the division by x + 1 gives
2x³ - 5x² - x + 6 = (x + 1)(2x² - 7x + 6)
=(x + 1)(2x - 3)(x - 2)

5. replacing x by -1, the result must be 0 :
-a² - a + 2 = 0 then a = 1 or a = -2

6. (0 ; 0), (1 ; 0) and (-2 ; 0)

7. See the graph !

8. With the graph : 4 roots

9. no rationnal root

10. if x = -3 , 3x² + 2x + m = 27 - 6 + m = 0
m = -21