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 for the problems using the Factor Theorem. Question 1: তলৰ কোনটো বহুপদৰ এটা উৎপাদক (x+1) তাক নিৰ্ণয় কৰা। (Determine which of the following polynomials has (x+1) as a factor.) According to the Factor Theorem, (x+1) is a factor of a polynomial p(x) if p(-1) = 0. a) p(x) = x^3 + x^2 + x + 1 Step 1: Substitute x = -1 into the polynomial. p(-1) = (-1)^3 + (-1)^2 + (-1) + 1 Step 2: Simplify the expression. p(-1) = -1 + 1 - 1 + 1 = 0 Since p(-1) = 0, (x+1) is a factor of x^3 + x^2 + x + 1. b) p(x) = x^4 + x^3 + x^2 + x + 1 Step 1: Substitute x = -1 into the polynomial. p(-1) = (-1)^4 + (-1)^3 + (-1)^2 + (-1) + 1 Step 2: Simplify the expression. p(-1) = 1 - 1 + 1 - 1 + 1 = 1 Since p(-1) ≠ 0, (x+1) is not a factor of x^4 + x^3 + x^2 + x + 1. c) p(x) = x^4 + 3x^3 + 3x^2 + x + 1 Step 1: Substitute x = -1 into the polynomial. p(-1) = (-1)^4 + 3(-1)^3 + 3(-1)^2 + (-1) + 1 Step 2: Simplify the expression. p(-1) = 1 + 3(-1) + 3(1) - 1 + 1 p(-1) = 1 - 3 + 3 - 1 + 1 = 1 Since p(-1) ≠ 0, (x+1) is not a factor of x^4 + 3x^3 + 3x^2 + x + 1. d) p(x) = x^3 - x^2 - (2+sqrt(2))x + sqrt(2) Step 1: Substitute x = -1 into the polynomial. p(-1) = (-1)^3 - (-1)^2 - (2+sqrt(2))(-1) + sqrt(2) Step 2: Simplify the expression. p(-1) = -1 - 1 + (2+sqrt(2)) + sqrt(2) p(-1) = -2 + 2 + sqrt(2) + sqrt(2) = 2sqrt(2) Since p(-1) ≠ 0, (x+1) is not a factor of x^3 - x^2 - (2+sqrt(2))x + sqrt(2). Therefore, only polynomial (i) x^3 + x^2 + x + 1 has (x+1) as a factor. Question 2: উৎপাদক উপপাদ্য ব্যৱহাৰ কৰি তলৰ প্ৰতিটো ক্ষেত্ৰতে g(x), p(x) ৰ এটা উৎপাদক হয়নে নহয় পৰীক্ষা কৰা: (Using the Factor Theorem, check whether g(x) is a factor of p(x) in each of the following cases:) a) p(x) = 2x^3 + x^2 - 2x - 1, g(x) = x + 1 Step 1: For g(x) = x+1 to be a factor, we must check if p(-1) = 0. p(-1) = 2(-1)^3 + (-1)^2 - 2(-1) - 1 Step 2: Simplify the expression. p(-1) = 2(-1) + 1 - (-2) - 1 p(-1) = -2 + 1 + 2 - 1 = 0 Since p(-1) = 0, g(x) is a factor of p(x). b) p(x) = x^3 + 3x^2 + 3x + 1, g(x) = x + 2 Step 1: For g(x) = x+2 to be a factor, we must check if p(-2) = 0. p(-2) = (-2)^3 + 3(-2)^2 + 3(-2) + 1 Step 2: Simplify the expression. p(-2) = -8 + 3(4) - 6 + 1 p(-2) = -8 + 12 - 6 + 1 p(-2) = 4 - 6 + 1 = -1 Since p(-2) ≠ 0, g(x) is not a factor of p(x). c) p(x) = x^3 - 4x^2 + x + 6, g(x) = x - 3 Step 1: For g(x) = x-3 to be a factor, we must check if p(3) = 0. p(3) = (3)^3 - 4(3)^2 + 3 + 6 Step 2: Simplify the expression. p(3) = 27 - 4(9) + 3 + 6 p(3) = 27 - 36 + 3 + 6 p(3) = -9 + 3 + 6 = 0 Since p(3) = 0, g(x) is a factor of p(x). Question 3: যদি (x-1), p(x) ৰ এটা উৎপাদক হয়, তেন্তে তলৰ প্ৰতিটো ক্ষেত্ৰতে k ৰ মান নিৰ্ণয় কৰা: (If (x-1) is a factor of p(x), then find the value of k in each of the following cases:) The expressions for p(x) containing k are not visible in the provided image. Please provide the polynomials for Question 3. That's 2 down. 3 left today — send the next one.