SOLUTION: True or False? A number k is a root of P(x) if and only if the remainder, when dividing P(x) by x+k, equals zero.

Question 1195036: True or False?

A number k is a root of P(x) if and only if the remainder, when dividing P(x) by x+k, equals zero.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: False

Why is this? Because the x+k should be x-k
The remainder of P(x)/(x-k) is P(k)

If the remainder is 0, then (x-k) is a factor of P(x)
We can say
P(x) = (x-k)Q(x)
where Q is the quotient polynomial.

Further Reading about the remainder theorem can be found here
https://www.purplemath.com/modules/remaindr.htm
https://www.mathsisfun.com/algebra/polynomials-remainder-factor.html

Answer by ikleyn(52855)   (Show Source): You can put this solution on YOUR website!
.
True or False?

A number k is a root of P(x) if and only if the remainder, when dividing P(x) by x+k, equals zero.
~~~~~~~~~~~~~~~~~~~~~~

As written in the post, the statement is FALSE.

The correct statement is THIS

A number k is a root of P(x) if and only if the remainder, when dividing P(x) by x-k, equals zero.

        ***********************************
                It is the Remainder Theorem.
        ***********************************

   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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