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
Okay, I will continue with the modular arithmetic problems, ensuring all math is in LaTeX and following the specified format. 9) Evaluate (3x+6)(3x-2) in mod 9 Step 1: Expand the product. (3x+6)(3x-2) = (3x)(3x) + (3x)(-2) + (6)(3x) + (6)(-2) = 9x^2 - 6x + 18x - 12 = 9x^2 + 12x - 12 Step 2: Reduce the coefficients modulo 9. 9x^2 0x^2 0 9 12x 3x 9 (since 12 = 1 × 9 + 3) -12 6 9 (since -12 = -2 × 9 + 6 or -12 + 18 = 6) Step 3: Substitute the reduced coefficients back into the expression. 9x^2 + 12x - 12 0 + 3x + 6 9 The expression in mod 9 is 3x + 6. 10) Evaluate (x^2+2x+1) ÷ (x+1) in mod 5 Step 1: Reduce the coefficients modulo 5. The coefficients in x^2+2x+1 and x+1 are already in their smallest non-negative residues modulo 5. Step 2: Perform polynomial division modulo 5. We can recognize that the numerator is a perfect square: x^2+2x+1 = (x+1)^2. So, the division becomes: ((x+1)^2)/(x+1) = x+1 Alternatively, using polynomial long division modulo 5: ` x + 1 _________ x + 1 | x^2 + 2x + 1 -(x^2 + x) ___________ x + 1 -(x + 1) _________ 0 ` The quotient is x+1 with a remainder of 0. The evaluated expression in mod 5 is x+1.