Using Descartes' Rule of Signs, determine the number of real solutions to 4x^{7} + 3x^{6} + x^{5} + 2x^{4} - x^{3} + 9x^{2} + x + 1

We first evaluate the possible positive roots using ƒ(x) = 4x^{7} + 3x^{6} + x^{5} + 2x^{4} - x^{3} + 9x^{2} + x + 1 There are 2 sign change(s) detailed below: Sign Change 1) + to - Sign Change 2) - to +

To find the remaining possible positive roots, we count down in pairs until we pass zero. 2 roots - 1 pair (2 roots) = 0 Therefore, we have a possible combination of (2 or 0) positive roots

Calculate possible negative roots: Given ƒ(x) = 4x^{7} + 3x^{6} + x^{5} + 2x^{4} - x^{3} + 9x^{2} + x + 1, we first need to determine ƒ(-x) ƒ(-x) = 4(-x)^{7} + 3(-x)^{6} + (-x)^{5} + 2(-x)^{4} - (-x)^{3} + 9(-x)^{2} + (-x) + 1

-x raised to an even power is positive. Odd exponents become negative: 4(-x)^{7} has a positive constant and odd exponent for a negative result of -4x^{7} 3(-x)^{6} has a positive constant and even exponent for a positive result of + 3x^{6} (-x)^{5} has a positive constant and odd exponent for a negative result of - x^{5} 2(-x)^{4} has a positive constant and even exponent for a positive result of + 2x^{4} -(-x)^{3} has a negative constant and odd exponent for a positive result of + x^{3} 9(-x)^{2} has a positive constant and even exponent for a positive result of + 9x^{2} (-x) has a positive constant and odd exponent for a negative result of - x 1 has a positive constant and even exponent for a positive result of + 1 ƒ(-x) = -4x^{7} + 3x^{6} - x^{5} + 2x^{4} + x^{3} + 9x^{2} - x + 1

We first evaluate the possible negative roots using ƒ(x) = - 4x^{7} + 3x^{6} - x^{5} + 2x^{4} + x^{3} + 9x^{2} - x + 1 There are 5 sign change(s) detailed below: Sign Change 1) - to + Sign Change 2) + to - Sign Change 3) - to + Sign Change 4) + to - Sign Change 5) - to +

To find the remaining possible negative roots, we count down in pairs until we pass zero. 5 roots - 1 pair (2 roots) = 3 3 roots - 1 pair (2 roots) = 1 Therefore, we have a possible combination of (5 or 3 or 1) negative roots