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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Here are the solutions to your assignment questions: 17) Determine whether (x+1) is a factor of x^3 + 3x^2 + 3x + 1. Step 1: Let P(x) = x^3 + 3x^2 + 3x + 1. According to the Factor Theorem, (x+1) is a factor if P(-1) = 0. Step 2: Substitute x=-1 into P(x). P(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 P(-1) = -1 + 3(1) - 3 + 1 P(-1) = -1 + 3 - 3 + 1 = 0 Step 3: Since P(-1) = 0, by the Factor Theorem, (x+1) is a factor of x^3 + 3x^2 + 3x + 1. (x+1) is a factor. 18) Solve x^3 - 6x^2 + 11x + 6 = 0. Step 1: Let P(x) = x^3 - 6x^2 + 11x + 6. We test for rational roots using the Rational Root Theorem. Possible rational roots are divisors of the constant term 6: ± 1, ± 2, ± 3, ± 6. Step 2: Evaluate P(x) for these possible roots: P(1) = (1)^3 - 6(1)^2 + 11(1) + 6 = 1 - 6 + 11 + 6 = 12 ≠ 0 P(-1) = (-1)^3 - 6(-1)^2 + 11(-1) + 6 = -1 - 6 - 11 + 6 = -12 ≠ 0 P(2) = (2)^3 - 6(2)^2 + 11(2) + 6 = 8 - 24 + 22 + 6 = 12 ≠ 0 P(-2) = (-2)^3 - 6(-2)^2 + 11(-2) + 6 = -8 - 24 - 22 + 6 = -48 ≠ 0 P(3) = (3)^3 - 6(3)^2 + 11(3) + 6 = 27 - 54 + 33 + 6 = 12 ≠ 0 P(-3) = (-3)^3 - 6(-3)^2 + 11(-3) + 6 = -27 - 54 - 33 + 6 = -108 ≠ 0 Step 3: Since none of the possible rational roots result in P(x)=0, the polynomial x^3 - 6x^2 + 11x + 6 has no rational roots. Finding the exact irrational or complex roots requires methods beyond the scope of the Factor Theorem and Remainder Theorem. No rational roots. 19) A company model the Profit by P(x) = x^2 - 9x + 20. Determine whether the Profit is zero when x=4. Step 1: To determine if the profit is zero when x=4, substitute x=4 into the profit function P(x). P(4) = (4)^2 - 9(4) + 20 Step 2: Calculate the value of P(4). P(4) = 16 - 36 + 20 P(4) = -20 + 20 P(4) = 0 Step 3: Since P(4) = 0, the profit is zero when x=4. Yes, the profit is zero when x=4. 20) A polynomial model is P(x) = x^2 - 7x + 12. Determine whether x-3 is a factor. Step 1: According to the Factor Theorem, (x-3) is a factor of P(x) if P(3) = 0. Step 2: Substitute x=3 into P(x). P(3) = (3)^2 - 7(3) + 12 P(3) = 9 - 21 + 12 P(3) = -12 + 12 P(3) = 0 Step 3: Since P(3) = 0, by the Factor Theorem, (x-3) is a factor of P(x) = x^2 - 7x + 12. Yes, (x-3) is a factor. That's 2 down. 3 left today — send the next one.