If
x 1
is a factor of
2 3 1
3 2
x kx x
, what is the value of k?
To determine the value of k, we need to find the remainder when
2 3 1
3 2
x - kx + x
is divided by (x - 1).
Using polynomial division, we have:
2x^2 - kx + x
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x - 1 | 2x^3 + 3x^2 - kx + x
-(2x^3 - 2x^2) (subtract previous term multiplied by divisor)
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5x^2 - kx + x
-(5x^2 - 5x) (subtract previous term multiplied by divisor)
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4x
The remainder is 4x.
Since the remainder is zero when x - 1 is a factor, the value of k must make the remainder zero. Therefore:
4x = 0
Solving for x, we get:
x = 0
Since x = 0, we can substitute this value into the original equation to find k:
2(0) - k(0) + 0 = 0
0 - 0 + 0 = 0
Therefore, k can take any value and the expression x - kx + x will still have x - 1 as a factor.
To solve this problem, we need to apply the factor theorem. According to the factor theorem, if 𝑥-1 is a factor of the given polynomial, then the polynomial evaluated at 𝑥=1 should be equal to zero.
So, let's substitute 𝑥=1 into the given polynomial:
(2(1) - k(1) + 1)(3(1) - 2) = 0
(2 - k + 1)(3 - 2) = 0
(3 - k)(1) = 0
Now, if the product of two numbers is zero, then at least one of the numbers must be zero. So we have:
3 - k = 0
Solving this equation for 𝑘:
-k = -3
𝑘 = 3
Therefore, the value of 𝑘 is 3.
To find the value of k, we need to use the factor theorem. The factor theorem states that if a polynomial f(x) is divided by a factor x - a and the remainder is zero, then (x - a) is a factor of f(x).
In this case, we are given that (x - 1) is a factor of the polynomial:
2x^3 - kx^2 + x + 3.
When (x - 1) is a factor, it means that when we substitute x = 1 into the polynomial, the result should be zero.
Let's substitute x = 1 into the polynomial:
2(1)^3 - k(1)^2 + 1 + 3
= 2 - k + 1 + 3
= 6 - k.
Now, we know that the result should be zero, so we can set it equal to zero:
6 - k = 0.
To solve for k, we can rearrange the equation to isolate k:
k = 6.
Therefore, the value of k is 6.