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Sagot :
We are asked to divide the following polynomials:
[tex]\frac{3x^3+7x^2-5x+8}{x^2+2x+4}[/tex]To do that we proceed like this:
[tex]3x^3+7x^2-5x+8\parallel x^2+2x+4[/tex]now we need to find a term that when multiplied by the first term on the right gives us the first term on the left, that term is 3x, because:
[tex](x^2)(3x)=3x^3[/tex]so, we multiply every term on the right by 3x, and substract that to the polynomial on the left, like this:
[tex]3x^3+7x^2-5x+8-(3x^3+6x^2+12x)[/tex]Solving the operations:
[tex]3x^3+7x^2-5x+8-3x^3-6x^2-12x=x^2-17x+8[/tex]Now we repeat this proceedure for the polynomial we got, like this:
[tex]x^2-17x+8\parallel x^2+2x+4[/tex]we need to multiply by 1, since
[tex]x^2(1)=x^2[/tex]multiplying by the polynomial on the right and subtracting:
[tex]x^2-17x+8-(x^2+2x+4)[/tex]Solving the operations:
[tex]x^2-17x+8-x^2-2x-4=-19x+4[/tex]Since the maximum exponent of the reulting polynomial is smaller than the maximum exponen of the polynomial we are dividing, we can't divide any more. The result is we take the last polynomial we got, and divide it by the polynomial we are dividing, and add the sum of the terms we were multiplying, that is 3x + 1, like this:
[tex]\frac{3x^3+7x^2-5x+8}{x^2+2x+4}=\frac{-19x+4}{x^2+2x+4}+3x+1[/tex]
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