Express f(x) in the form f(x)=(x-k)q(x)+r for the given value 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 long division.
First, let's set up the long division:
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(x - k) | x^3 + 6x^2 + 10x + 4
The divisor is x - k, where k = -2, so we have:
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(x + 2) | x^3 + 6x^2 + 10x + 4
To start the long division, we divide the highest degree term of the dividend (x^3) by the highest degree term of the divisor (x), which gives us x^2. We write this above the division line:
x^2
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(x + 2) | x^3 + 6x^2 + 10x + 4
Then, we multiply the divisor (x + 2) by the quotient (x^2), which gives us x^3 + 2x^2. We place this below the dividend and subtract it:
x^2
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(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
This simplifies to:
x^2
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(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x
Now, we bring down the next term from the dividend, which is 4x^2:
x^2 + <<4x^2>> (bring down 4x^2)
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(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x
Next, we divide the highest remaining degree term in the new dividend (4x^2) by the highest degree term of the divisor (x), which gives us 4x. We write this above the division line:
x^2 + 4x
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(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x
Now, we multiply the divisor (x + 2) by the new quotient (4x), which gives us 4x^2 + 8x. We place this below the dividend and subtract it:
x^2 + 4x
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(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x
- (4x^2 + 8x)
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2x + 4
We bring down the next term from the dividend, which is 2x:
x^2 + 4x + <<2x>> (bring down 2x)
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(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x
- (4x^2 + 8x)
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2x + 4
Finally, we divide the remaining term in the new dividend (2x) by the divisor (x), which gives us 2. We write this above the division line:
x^2 + 4x + 2
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(x + 2) | x^3 + 6x^2 + 10x + 4
- (x^3 + 2x^2)
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4x^2 + 10x
- (4x^2 + 8x)
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2x + 4
At this point, we have fully divided the dividend by the divisor, and there is no remainder. Therefore, the expression f(x) can be written as:
f(x) = (x + 2)(x^2 + 4x + 2)
And that is the expression for f(x) in the intended form given k = -2.
So, f(x) = (x + 2)(x^2 + 4x + 2).