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Question 1170225: the binomial (x+4) is a factor of the following polynomials, find the value of k.
1.2x^3-3x^2-kx+60
2.7x^2+kx-12
3.kx^3-10x^2-137x+60
4.4x^2+kx-4
5.8x^3+kx^2-53x+12
Find the values of A, B and C if (x-1),(x+2), and (x-2) are factors of
2Ax^4-Bx^3-Cx-16
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52783)   (Show Source):
You can put this solution on YOUR website! .
In problems 1 - 5 below, the idea of the solution is the same.
You substitute the value of the root x= -4 into the formula and equate this expression to zero.
It gives you an equation to find "k".
Having this idea explained to you, you can complete the rest on your own.
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Comment from student: what can you answer number 1 so I know what to do to others?
My response :
It is REALLY GOOD IDEA start working with one specific case, instead of posting 5 same type problems in one post.
1. the binomial (x+4) is a factor of the following polynomials, find the value of k.
2x^3 - 3x^2 - kx + 60
~~~~~~~~~~~~~~~
Since (x+4) is the root of the given polynomial, it means, due to the Remainder Theorem,
that (-4) is the root of the polynomial.
So, we substitute x= -4 into the polynomial and equate this expression to zero (since -4 is the root (!) )
We get then
2*(-4)^3 - 3*(-4)^2 - k*(-4) + 60 = 0.
It is our BASIC EQUATION to find the unknown value of "k" :
2*(-64) - 3*(16) - k*(-4) + 60 = 0
-128 - 48 + 4k + 60 = 0
-116 + 4k = 0
4k = 116.
k = 116/4 = 29.
ANSWER. k = 29.
Solved.
Do other cases IN THE SAME WAY.
/\/\/\/\/\/\/\/
It is a good problem, and it is my pleasure to teach you.
Do you understand everything in my solution ?
If you have questions, do not hesitate to post them to me.
/\/\/\/\/\/\/\/
Never submit many problems/questions of the same type in one post.
It is really irritates tutors; works against you and creates bad impression about a visitor.
By the way, it is PROHIBITED by the rules of this forum.
Happy learning (!)
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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.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website! the binomial (x+4) is a factor of the following polynomials, find the value of k.
1.2x^3-3x^2-kx+60
2.7x^2+kx-12
3.kx^3-10x^2-137x+60
4.4x^2+kx-4
5.8x^3+kx^2-53x+12
Find the values of A, B and C if (x-1),(x+2), and (x-2) are factors of
2Ax^4-Bx^3-Cx-16
With a factor of the polynomial being x + 4, a zero is - 4.
We then get: 
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