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Answer (1)
Answer:The problem is to divide (2x^3 + 9x^2 + 3x - 4) by (x + 4).We'll use polynomial long division to find the quotient and remainder. * Set up the division: Write the problem in the standard long division format. * Divide the leading terms: Divide the first term of the dividend (2x^3) by the first term of the divisor (x). The result is 2x^2. This is the first term of your quotient. (2x^3 / x) = 2x^2 Now, multiply the entire divisor (x + 4) by 2x^2: 2x^2(x + 4) = 2x^3 + 8x^2 * Subtract and bring down: Subtract this result from the first part of the dividend: (2x^3 + 9x^2) - (2x^3 + 8x^2) = x^2 Bring down the next term from the dividend, which is 3x: x^2 + 3x * Repeat the process: Divide the new leading term (x^2) by the divisor's leading term (x): (x^2 / x) = x This is the next term of your quotient. Multiply the divisor by x: x(x + 4) = x^2 + 4x Subtract this from x^2 + 3x: (x^2 + 3x) - (x^2 + 4x) = -x Bring down the last term from the dividend, which is -4: -x - 4 * Final step: Divide the new leading term (-x) by the divisor's leading term (x): (-x / x) = -1 This is the final term of your quotient. Multiply the divisor by -1: -1(x + 4) = -x - 4 Subtract this from -x - 4: (-x - 4) - (-x - 4) = 0 * The Result: The quotient is 2x^2 + x - 1 and the remainder is 0.The final answer in the form P(x) = Q(x) + \frac{R(x)}{d(x)} is:2x^3 + 9x^2 + 3x - 4 = (2x^2 + x - 1) + \frac{0}{x + 4}
