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) (Show Source):
You can put this solution on YOUR website! .
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 = .
Then from (4), c = 2 - b = 2 - = .
So, the polynomial is f(x) = . 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 by the binomial is equal to the value of the polynomial.
2. The binomial divides the polynomial if and only if the value of is the root of the polynomial , i.e. .
3. The binomial factors the polynomial if and only if the value of is the root of the polynomial , i.e. .
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.
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