Question

Find the mistake in the question and explain. show work with correct answer

Answer 1

5. The correct answer is P(-2) = 316.

6. There are no real roots for the given polynomial
`\(f(x) = x^4 - 5x^2 - 36\)`.

How did we arrive at these assertions?

Problem 5: The mistake in the question is in the synthetic division. The coefficients in the synthetic division should be multiplied by the divisor, but it is subtracted directly.

The correct synthetic division should be:

1 -47 -16 8 52

[-2] -2 98 -164 312

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1 -49 82 -156 (316)

Therefore, the remainder is 316, not 364. So, the correct answer is P(-2) = 316.

Problem 6: The mistake in the question is in the synthetic division as well.

The correct synthetic division should be:

1 0 -5 0 -36

[-3] -3 9 -12 36

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1 -3 4 -12 0

[3] 3 0 12

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1 0 4 0

The quotient is
`\(x^2 + 4 = 0\)`, which leads to
`\(x^2 = -4\)`. Taking the square root of both sides gives
`\(x = \pm 2i\)` for the imaginary roots, but there are no real roots from this equation.

Therefore, the correct conclusion is that there are no real roots for the given polynomial
`\(f(x) = x^4 - 5x^2 - 36\)`.