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
Is f(x) =0 ?
To express f(x) in the given form, we need to perform polynomial long division using k=-2.
First, let's set up the polynomial long division:
x + 4
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x + 2 | x^3 + 6x^2 + 10x + 4
We divide the first term, x^3, by x, which gives us x^2. We then multiply (x + 2) by x^2 and subtract it from the given polynomial:
x^2(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 + 4
We repeat the process with the new polynomial 4x^2 + 10x + 4. We divide the first term, 4x^2, by x, which gives us 4x. We then multiply (x + 2) by 4x and subtract it from the new polynomial:
4x(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 + 4
-(4x^2 + 8x)
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2x + 4
Finally, we have the remainder 2x + 4. Thus, we can express f(x) in the desired form:
f(x) = (x + 2)(x^2 + 4x + 2) + (2x + 4)
Now, to answer the question "Is f(x) = 0?", we need to substitute f(x) into the equation and solve for x:
(x + 2)(x^2 + 4x + 2) + (2x + 4) = 0
This is a quadratic equation, and to solve it, we can either factor it or use the quadratic formula. Once we have the solutions for x, we can determine if f(x) equals zero for any of those solutions.