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
Step 1: Find a root of the polynomial P(z) = z^3 - 6z^2 + 13z - 10 using the Rational Root Theorem. The possible rational roots are the divisors of the constant term (-10), which are ± 1, ± 2, ± 5, ± 10. Let's test z=2: P(2) = (2)^3 - 6(2)^2 + 13(2) - 10 P(2) = 8 - 6(4) + 26 - 10 P(2) = 8 - 24 + 26 - 10 P(2) = 34 - 34 = 0 Since P(2)=0, z=2 is a root, and (z-2) is a linear factor. Step 2: Divide the polynomial P(z) by (z-2) to find the quadratic factor. We can use synthetic division: ` 2 | 1 -6 13 -10 | 2 -8 10 ----------------- 1 -4 5 0 ` The quotient is z^2 - 4z + 5. So, z^3 - 6z^2 + 13z - 10 = (z-2)(z^2 - 4z + 5). Step 3: Factor the quadratic term z^2 - 4z + 5. We use the quadratic formula z = -b ± sqrt(b^2 - 4ac)2a for az^2 + bz + c = 0. Here, a=1, b=-4, c=5. Calculate the discriminant = b^2 - 4ac: = (-4)^2 - 4(1)(5) = 16 - 20 = -4 Now, find the roots: z = -(-4) ± sqrt(-4)2(1) z = (4 ± 2i)/(2) z = 2 ± i The two complex roots are z_1 = 2+i and z_2 = 2-i. Therefore, the linear factors for the quadratic term are (z - (2+i)) and (z - (2-i)). Step 4: Write the complete factorization into linear factors. The linear factors are (z-2), (z-(2+i)), and (z-(2-i)). z^3 - 6z^2 + 13z - 10 = (z-2)(z-(2+i))(z-(2-i)) z^3 - 6z^2 + 13z - 10 = (z-2)(z-2-i)(z-2+i) The factorization of z^3 - 6z^2 + 13z - 10 = 0 into linear factors is: (z-2)(z-2-i)(z-2+i) Send me the next one 📸