The binomial x-3 divides exactly the polynomial x^3-2x^2+cx-12 and c is a whole number then c is what?
Review how to do polynomial long division and give it a try. You will find that you need a value of c such that
(c+3) x -12 is evenly divisible by x-3.
See http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Synthetic%20Division.html
To find the value of c, we need to divide the polynomial x^3 - 2x^2 + cx - 12 by x - 3, and determine if the division is exact.
We can do this using polynomial long division. Here are the steps:
Step 1: Set up the division:
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x - 3 | x^3 - 2x^2 + cx - 12
Step 2: Divide the first term of the dividend by the first term of the divisor:
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x - 3 | x^3 - 2x^2 + cx - 12
x^(3-1) = x^2
Step 3: Multiply the divisor (x - 3) by the result obtained in Step 2, then write the result below the dividend:
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x - 3 | x^3 - 2x^2 + cx - 12
-(x^3 - 3x^2)
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x^2 + cx
Step 4: Subtract the result obtained in Step 3 from the dividend:
___________
x - 3 | x^3 - 2x^2 + cx - 12
-(x^3 - 3x^2)
______________
x^2 + cx - 12
Step 5: Repeat Steps 2-4 with the new dividend (x^2 + cx - 12).
We need to ensure that there is no remainder after the division for it to be exact. In this case, c is a whole number, which means the coefficient of x in the dividend should also be a whole number.
So, let's continue:
Step 2:
Divide the first term of the new dividend (x^2) by the first term of the divisor (x):
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x - 3 | x^2 + cx - 12
x^(2-1) = x
Step 3:
Multiply the divisor (x - 3) by the result obtained in Step 2:
______________
x - 3 | x^2 + cx - 12
-(x^2 - 3x)
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cx - 12
Step 4:
Subtract the result obtained in Step 3 from the new dividend:
______________
x - 3 | x^2 + cx - 12
-(x^2 - 3x)
_____________
cx - 12 + 3x - (-12)
= cx + 3x
Now, we check if there is any remainder. Since we are dividing by x - 3, if the remainder is 0, the division is exact.
To have a zero remainder, the coefficient of x in the new dividend (cx + 3x) must be zero. Therefore, we have:
cx + 3x = 0
(c + 3) x = 0
For the division to be exact, the coefficient of x must be zero. Therefore, c + 3 = 0. Solving for c:
c + 3 = 0
c = -3
So, if c is a whole number and the binomial x - 3 divides the polynomial x^3 - 2x^2 + cx - 12 exactly, then c is equal to -3.