(2×⁴-2׳-15ײ+9×-6)÷(×-3)
Answer (1)
Answer:To solve the division \((2x^4 - 2x^3 - 15x^2 + 9x - 6) \div (x - 3)\), we can use polynomial long division. Let's go through the steps:### Step 1: Divide the first term of the dividend by the first term of the divisor.- Divide \(2x^4\) by \(x\): \(2x^3\).### Step 2: Multiply the entire divisor by the result from Step 1.- \(2x^3 \times (x - 3) = 2x^4 - 6x^3\).### Step 3: Subtract this product from the original polynomial.- \[(2x^4 - 2x^3 - 15x^2 + 9x - 6) - (2x^4 - 6x^3)\]- This gives: \(4x^3 - 15x^2 + 9x - 6\).### Step 4: Repeat the process with the new polynomial.- Divide \(4x^3\) by \(x\): \(4x^2\).- Multiply \(4x^2\) by the divisor \((x - 3)\): \(4x^3 - 12x^2\).- Subtract: \((4x^3 - 15x^2 + 9x - 6) - (4x^3 - 12x^2)\): - This gives: \(-3x^2 + 9x - 6\).### Step 5: Repeat again.- Divide \(-3x^2\) by \(x\): \(-3x\).- Multiply \(-3x\) by the divisor \((x - 3)\): \(-3x^2 + 9x\).- Subtract: \((-3x^2 + 9x - 6) - (-3x^2 + 9x)\):- This gives: \(-6\).### Step 6: Final division.- Divide \(-6\) by \(x\): \(-6/x\).- Since this is a constant and \(x-3\) is a linear polynomial, the division ends here.### Answer:The quotient is \(2x^3 + 4x^2 - 3x + 0\) and the remainder is \(-6\).So, the final result can be written as:\[2x^3 + 4x^2 - 3x + \frac{-6}{x-3}.\]
