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's the solution for Question Q3: To simplify the expression (6x^2 + 7x - 5)/(2x - 1) by dividing out, we can use polynomial long division. Step 1: Set up the polynomial long division. c|cc cc 2r3x & +5 \\ 2-5 2x-1 & 6x^2 & +7x & -5 \\ 2r-(6x^2 & -3x) \\ 2-3 2r0 & 10x & -5 \\ 2r & -(10x & -5) \\ 3-4 2r & 0 & 0 \\ Step 2: Divide the first term of the dividend (6x^2) by the first term of the divisor (2x). (6x^2)/(2x) = 3x Write 3x as the first term of the quotient. Step 3: Multiply the divisor (2x - 1) by 3x. 3x(2x - 1) = 6x^2 - 3x Subtract this result from the dividend. (6x^2 + 7x) - (6x^2 - 3x) = 10x Bring down the next term, -5. The new dividend is 10x - 5. Step 4: Divide the first term of the new dividend (10x) by the first term of the divisor (2x). (10x)/(2x) = 5 Write 5 as the next term of the quotient. Step 5: Multiply the divisor (2x - 1) by 5. 5(2x - 1) = 10x - 5 Subtract this result from the new dividend. (10x - 5) - (10x - 5) = 0 The remainder is 0. The quotient is 3x + 5. Alternatively, by factoring the numerator: We look for two numbers that multiply to 6 × -5 = -30 and add to 7. These numbers are 10 and -3. 6x^2 + 7x - 5 = 6x^2 + 10x - 3x - 5 = 2x(3x + 5) - 1(3x + 5) = (2x - 1)(3x + 5) Now, substitute this back into the fraction: ((2x - 1)(3x + 5))/(2x - 1) Assuming 2x - 1 ≠ 0, we can cancel out the common factor (2x - 1). = 3x + 5 The simplified expression is 3x + 5 Send me the next one 📸