Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use synthetic division to divide. $$\frac{4 x^{3}+16 x^{2}-23 x-15}{x+\frac{1}{2}}$$

Step-by-Step Solution

Answer

By applying synthetic division, the quotient when \(4x^{3} + 16x^{2} - 23x - 15\) is divided by \(x + \frac{1}{2}\) is \(4x^{2} + 14x -16 - \frac{7}{x + \frac{1}{2}}\).

Step 1: Identify the coefficients and root

From the dividend \(4x^{3} + 16x^{2} - 23x - 15\), the coefficients are [4, 16, -23, -15]. The root from the divisor \(x + \frac{1}{2}\) is \(-\frac{1}{2}\).

Step 2: Start synthetic division

Write the coefficients and root as such: \(-\frac{1}{2}\) | 4 16 -23 -15. Bring down the first coefficient (4) unchanged as part of the quotient.

Step 3: Continue synthetic division

Multiply the root (\(-\frac{1}{2}\)) by the first quotient (4) and place the result under the second coefficient (16) and add to get the new quotient coefficient. Repeat this for all coefficients.

Step 4: Identify the resulting quotient

The final row of numbers represents the coefficients of the quotient polynomial. The degree of each term reduces by 1, so the quotient polynomial from this division is \(4x^{2} + 14x -16\). The final number is the remainder, -7, so the complete quotient is \(4x^{2} + 14x -16 - \frac{7}{x + \frac{1}{2}}\).