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the simplified form of (x^4 - x^2 - 7)/(x + 4) is x^3 - 4x -15 . REMAINDER -67
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The expression (x^4 - x^2 - 7)/(x + 4) represents a rational function. To simplify it, we can use polynomial long division or synthetic division.
Let's use polynomial long division:
Step 1: Divide the highest degree term of the numerator (x^4) by the highest degree term of the denominator (x), which gives us x^3.
x^3
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x + 4 | x^4 - x^2 - 7
Step 2: Multiply the divisor (x + 4) by the quotient (x^3), and subtract the result from the numerator.
x^3(x + 4) = x^4 + 4x^3
(x^4 - x^2 - 7) - (x^4 + 4x^3) = -4x^3 - x^2 - 7
Step 3: Repeat the process with the new polynomial (-4x^3 - x^2 - 7).
-4x
___________
x + 4 | -4x^3 - x^2 - 7
Step 4: Multiply the divisor (x + 4) by the new quotient (-4x), and subtract the result from the remaining polynomial.
-4x(x + 4) = -4x^2 - 16x
(-4x^3 - x^2 - 7) - (-4x^2 - 16x) = 15x^2 + 15x - 7
Step 5: Repeat the process with the new polynomial (15x^2 + 15x - 7).
15
___________
x + 4 | 15x^2 + 15x - 7
Step 6: Multiply the divisor (x + 4) by the new quotient (15), and subtract the result from the remaining polynomial.
15(x + 4) = 15x + 60
(15x^2 + 15x - 7) - (15x + 60) = -67
Step 7: The remainder is -67.
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