Question

Find the real solutions of the following equation.

x^4 +7x^3+8x^2-7x+15=0

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

The solution set is { }

(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed. Type each answer only once.)

B.

The solution set is

empty set∅.

Answer 1

There are several ways of solving the equation x⁴ +7x³+8x²-7x+15=0

The easiest method is to prepare a graph of the function y = f(x) = x⁴ +7x³+8x²-7x+15. A graph of this function prepared by using a graphing calculator is attached. It may be observed that the graph crosses the X-Axis at only 2 points i.e. at x = -5 and x = -3. Hence, there are only 2 real solutions to the equation x⁴ +7x³+8x²-7x+15=0 which are x = -5 and x = -3.

The 2^nd method is to use the Rational Roots theorem. As per the rational roots theorem, if p is a factor of the constant term of a polynomial f(x) and if q is a factor of the coefficient of the leading term, then p/q is likely to be a root of f(x). Here, the factors of 15 are ± 1,3,5 and the factors of 1 are ± 1. Further, f(-5) = 0 and f(-3) = 0 so that -5 and -3 are real solutions to the equation x⁴ +7x³+8x²-7x+15=0. Further, on dividing f(x) = x⁴ +7x³+8x²-7x+15 by (x+5)(x+3) = x²+8x+15, we get x²-x+1. Hence, the remaining roots of f(x) are the roots of x²-x+1. On using the quadratic formula, these are [ 1±√(1-4)]/2 or, (1± i√3)/2 or, 1/2 ± (i√3)/2 or, 1/2 - (i√3)/2 and 1/2 + (i√3)/2. These are complex solutions to the equation x⁴ +7x³+8x²-7x+15=0.

Thus, the real solutions to the equation x⁴ +7x³+8x²-7x+15=0 are x = -5 and x = -3.

Option A.