Multiple choice

The equation $x^{4}+5x^{3}+5x^{2}-5x-6$ has two roots which are equal in magnitude and opposite in sign. They are

  1. $2, -2$
  2. $1, -1$
  3. $6, -6$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If roots are equal in magnitude and opposite in sign, they are a and -a. The sum of these two roots is 0. Testing the options, if 1 and -1 are roots, (x-1)(x+1) = x^2-1 must be a factor. Dividing the polynomial by x^2-1 yields x^2+5x+6 = (x+2)(x+3). The roots are 1, -1, -2, -3.

AI explanation

The polynomial x^4 + 5x^3 + 5x^2 - 5x - 6 can be factored by grouping into x^3(x + 5) + (x + 5)(x - 1). This simplifies to (x + 5)(x^3 + x - 1) = (x + 5)(x - 1)(x^2 + x + 1). Setting the factors to zero gives x = -5, x = 1, and complex roots, meaning the real roots equal in magnitude and opposite sign are 1 and -1.

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