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.
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Okay, let's continue with Question 1.2. Question 1.2: Use the factor theorem to find a linear factor of x^3 - 5x^2 + 7x - 3. Hence solve the equation x^3 - 5x^2 + 7x - 3 = 0. Step 1: Find a linear factor using the Factor Theorem. Let P(x) = x^3 - 5x^2 + 7x - 3. According to the Factor Theorem, if P(a) = 0, then (x-a) is a factor of P(x). We test integer factors of the constant term -3, which are ± 1, ± 3. Test x=1: P(1) = (1)^3 - 5(1)^2 + 7(1) - 3 P(1) = 1 - 5 + 7 - 3 P(1) = 8 - 8 P(1) = 0 Since P(1) = 0, (x-1) is a linear factor of P(x). Step 2: Perform polynomial division to find the quadratic factor. Divide x^3 - 5x^2 + 7x - 3 by (x-1). We can use synthetic division or long division. Using synthetic division with root 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: Solve the quadratic equation to find the remaining roots. We need to solve x^2 - 4x + 3 = 0. This quadratic equation can be factored: (x-1)(x-3) = 0 The roots from this quadratic factor are x-1=0 x=1 and x-3=0 x=3. Step 4: State all solutions for the equation. The solutions for x^3 - 5x^2 + 7x - 3 = 0 are the roots found from both the linear factor and the quadratic factor. The roots are x=1 (from the initial factor), x=1 (from the quadratic factor), and x=3. The linear factor is x-1 and the solutions to the equation are x=1, x=3. 3 done, 2 left today. You're making progress.