Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use synthetic division to divide. $$\left(3 x^{3}-17 x^{2}+15 x-25\right) \div(x-5)$$

Step-by-Step Solution

Answer

The quotient from the synthetic division of the polynomial \(3x^3 - 17x^2 + 15x - 25\) by the binomial \(x - 5\) is \(3x^2 - 2x + 5\).

Step 1: Set Up the Synthetic Division Problem

Arrange the coefficients of the polynomial to be divided (the dividend) and the divisor. Write down the coefficients of the polynomial in descending order of degrees. These numbers are 3 (from \(3x^3\)), -17 (from \(-17x^2\)), 15 (from \(15x\)), and -25 (from \(-25\)). Write them within a long division symbol. The number to put outside the long division symbol is the opposite of the constant in the binomial divisor, so -(-5) = 5 in this case.

Step 2: Perform the Synthetic Division

To start, drop the first coefficient (namely, 3) down. Then multiply this number by the one outside the division symbol (which is 5) and write the result beneath the second coefficient. Hence, 3 * 5 = 15. Write 15 beneath -17. Add -17 and 15 to get -2. Now, repeat this process: multiply -2 by 5 to get -10 and write this beneath 15. Add these to get 5. Again, multiply 5 by 5 to get 25 and write this beneath the -25. Add these to get 0.

Step 3: Interpret the Result

The numbers obtained inside the division symbol represent the coefficients of the quotient. Counting from the left, the first number, 3, is the coefficient of the \(x^2\) term because we started with an \(x^3\) term in the polynomial before division. The second number, -2, is the coefficient of the \(x\) term and the third number, 5, is the constant term. Given that the remainder is 0, the quotient of the division is \(3x^2 - 2x + 5\).