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
To solve Question 1.2, we will use the Factor Theorem and then factorize the polynomial. Step 1: Use the Factor Theorem to find a linear factor. Let f(x) = x^3 - 5x^2 + 7x - 3. According to the Factor Theorem, if (x-a) is a factor of f(x), then f(a) = 0. We test integer factors of the constant term, -3, which are ± 1, ± 3. Test x=1: f(1) = (1)^3 - 5(1)^2 + 7(1) - 3 f(1) = 1 - 5 + 7 - 3 f(1) = 8 - 8 f(1) = 0 Since f(1) = 0, (x-1) is a linear factor of f(x). Step 2: Divide the polynomial by the linear factor. We use synthetic division to divide x^3 - 5x^2 + 7x - 3 by (x-1). ` 1 | 1 -5 7 -3 | 1 -4 3 ---------------- 1 -4 3 0 ` The quotient is x^2 - 4x + 3. So, x^3 - 5x^2 + 7x - 3 = (x-1)(x^2 - 4x + 3). Step 3: Factorize the quadratic expression. Now we need to factorize the quadratic x^2 - 4x + 3. We look for two numbers that multiply to 3 and add to -4. These numbers are -1 and -3. So, x^2 - 4x + 3 = (x-1)(x-3). Step 4: Solve the equation. Substitute the factored quadratic back into the polynomial equation: (x-1)(x-1)(x-3) = 0 (x-1)^2(x-3) = 0 Set each factor to zero to find the solutions: x-1 = 0 x = 1 x-3 = 0 x = 3 The solutions to the equation x^3 - 5x^2 + 7x - 3 = 0 are x=1 (multiplicity 2), x=3. That's 2 down. 3 left today — send the next one.