SOLUTION: a. Find a polynomial of minimum degree such that when divided by x+2 has a remainder of -1 and when divided by x-1 has a remainder of 3. b. Find a polynomial of degree 3 such that

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: a. Find a polynomial of minimum degree such that when divided by x+2 has a remainder of -1 and when divided by x-1 has a remainder of 3. b. Find a polynomial of degree 3 such that      Log On


   



Question 1154812: a. Find a polynomial of minimum degree such that when divided by x+2 has a remainder of -1 and when divided by x-1 has a remainder of 3.
b. Find a polynomial of degree 3 such that when divided by x^2-5x has a remainder of 6x-15 and when divided by x^2-5x+8 has a remainder of 2x-7.

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
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            I will solve part  (a)  ONLY.


It is clear that the polynomial can not be linear (of the degree 1).


So, I will find such a polynomial of the degree 2 (quadratic).


Let f(x) = x^2 + bx + c be such a polynomial.


According to the Remainder theorem, the imposed conditions are equivalent to 

    f(-2) = -1  and  f(1) = 3,   or


    (-2)^2 - 2b + c = -1      (1)

    1^2    +  b + c =  3      (2)


Equations (1) and (2) are equivalent to


           - 2b + c = -5      (3)

              b + c =  2      (4)


From equation (3), subtract equation (4). You will get

            -3b     = -7;   hence,  b = 7%2F3.


Then from (4),  c = 2 - b = 2 - 7%2F3 = -1%2F3.


So, the polynomial is  f(x) =  x%5E2+%2B+%287%2F3%29x+-+1%2F3.    ANSWER


CHECK.  f(-2) = (-2)^2+(7/3)*(-2) - 1/3 = 4 - 14/3 - 1/3 = -1;

        f(1) = 1^2 + 7/3 - 1/3 = 1 + 2 = 3.    ! Correct !

Solved.

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   Theorem   (the remainder theorem)
   1. The remainder of division the polynomial  f%28x%29  by the binomial  x-a  is equal to the value  f%28a%29  of the polynomial.
   2. The binomial  x-a  divides the polynomial  f%28x%29  if and only if the value of  a  is the root of the polynomial  f%28x%29,  i.e.  f%28a%29+=+0.
   3. The binomial  x-a  factors the polynomial  f%28x%29  if and only if the value of  a  is the root of the polynomial  f%28x%29,  i.e.  f%28a%29+=+0.


See the lessons
    - Divisibility of polynomial f(x) by binomial x-a and the Remainder theorem
    - Solved problems on the Remainder thoerem
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.