Express f(x)in the form f(x)=(x-k)q(x)+r for the given values of k.
f(x)=x^3+6x^2+10x+4, k=-2
f(x)=____
To express f(x) in the form f(x) = (x - k)q(x) + r, we need to perform polynomial division. The value of k is given as -2.
First, we write the expression f(x) = x^3 + 6x^2 + 10x + 4 and set up the division:
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(x + 2) | x^3 + 6x^2 + 10x + 4
To start the division, we divide the first term x^3 by x, which gives us x^2. Then, we multiply x^2 by (x + 2) and subtract the result from the original expression:
x^2 + 2x
(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x
We continue the process by dividing 4x^2 by x, which gives us 4x. We then multiply 4x by (x + 2) and subtract the result:
x^2 + 2x - 4
(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x + 4
- (4x^2 + 8x)
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2x + 4
Finally, we divide 2x by x, which gives us 2. We multiply 2 by (x + 2) and subtract the result:
x^2 + 2x - 4
(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x + 4
- (4x^2 + 8x)
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2x + 4
- (2x + 4)
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0
Since there is no remainder (r = 0), we have expressed f(x) in the desired form.
Therefore, f(x) = (x + 2)(x^2 + 2x - 4) + 0.
f(x) = x^3 + 6x^2 + 10x + 4 can be expressed as f(x) = (x + 2)(x^2 + 2x - 4).