This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Step 1: Simplify the polynomial. The given polynomial is f(x) = x^3 + 4x + x - 6. Combine the like terms: f(x) = x^3 + 5x - 6 Step 2: Find a rational root using the Rational Root Theorem. We look for integer divisors of the constant term (-6), which are ± 1, ± 2, ± 3, ± 6. Let's test x=1: f(1) = (1)^3 + 5(1) - 6 = 1 + 5 - 6 = 0 Since f(1) = 0, x=1 is a zero of the polynomial, and (x-1) is a factor. Step 3: Use synthetic division to divide the polynomial by (x-1). Using synthetic division with the root 1: ` 1 | 1 0 5 -6 | 1 1 6 ---------------- 1 1 6 0 ` The quotient is x^2 + x + 6. So, f(x) = (x-1)(x^2 + x + 6). Step 4: Find the zeros of the quadratic factor x^2 + x + 6. Set the quadratic factor to zero: x^2 + x + 6 = 0. Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a, where a=1, b=1, c=6. The discriminant is = b^2 - 4ac = (1)^2 - 4(1)(6) = 1 - 24 = -23. Since the discriminant is negative, there are no real roots for this quadratic factor. The roots are complex: x = -1 ± sqrt(-23)2(1) x = -1 ± isqrt(23)2 Step 5: State all the zeros of the polynomial. The zeros of the polynomial are the real root found in Step 2 and the complex roots found in Step 4. The zeros are 1, -1 + isqrt(23)2, -1 - isqrt(23)2. Drop the next question.