Problem 20 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(-6 x^{3}+x^{2}-4\right) \div(2 x-3) $$

Step-by-Step Solution

Answer

Quotient is \(-6x^2 - 8x - 12\), remainder \(-22\).

Step 1: Set Up the Synthetic Division

To use synthetic division, we need to express the divisor and arrange the dividend appropriately. Let's set up the synthetic division process for \(-6x^3 + x^2 + 0x - 4\) divided by \(2x - 3\). The zero of the divisor \(2x - 3\) is \(\frac{3}{2}\) because setting \(2x - 3 = 0\) gives us \(x = \frac{3}{2}\). Place \(\frac{3}{2}\) in front and set up the polynomial coefficients: [-6, 1, 0, -4].

Step 2: Begin the Synthetic Division Process

Bring down the leading coefficient, which is \(-6\), as the starting value at the bottom row. This will be the first number of the quotient.

Step 3: Multiply and Add

1. Multiply \(-6\) (from the bottom row) by \(\frac{3}{2}\) (on the left) and write the resulting product in the next column of the middle row.2. Add this number to the coefficient in the upper row of the next column (in this case, \(1\)). Continue this process step-by-step for each coefficient.

Step 4: Complete the Process for Each Coefficient

Continue the process for each term:- Multiply the result from the bottom row by \(\frac{3}{2}\) and add the result to the next term in the top row.Perform this until you've worked through all coefficients of the polynomial: \[\begin{array}{cccc}\frac{3}{2} & | & -6 & 1 & 0 & -4 \ & | & & -9 & -12 & -18 \\hline & | & -6 & -8 & -12 & -22 \\end{array}\]The new bottom row \([-6, -8, -12, -22]\) represents the coefficients of the quotient polynomial and the remainder.

Step 5: Write the Quotient and the Remainder

The numbers \([-6, -8, -12]\) represent the coefficients of the polynomial and "-22" is the remainder. Thus, the quotient is:\[-6x^2 - 8x - 12\] with a remainder of \(-22\). This gives us\[ -6x^2 - 8x - 12 + \frac{-22}{2x-3} \].

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial, called the dividend, by another non-zero polynomial, called the divisor. It's similar to numerical long division. In this exercise, we used synthetic division, which simplifies the process of dividing polynomials when the divisor is in the form of a linear polynomial (like \(2x - 3\)).

Here are the basic steps involved in polynomial division through synthetic division:

• Identify the zero of the linear divisor by solving the equation \(2x - 3 = 0\).
• Use the zero, \(\frac{3}{2}\), and arrange the polynomial coefficients \([-6, 1, 0, -4]\) from highest to lowest degree.
• Apply the synthetic division process step-by-step to find the quotient polynomial and remainder.

It's a handy tool in algebra, simplifying complex division tasks without overwhelming calculations.

Quotient Polynomial

The quotient polynomial is the result obtained from dividing one polynomial by another. In our problem, the quotient we discovered using synthetic division is \(-6x^2 - 8x - 12\).

To find the quotient, synthetic division allows us to work with coefficients rather than the whole expression, making calculations less cumbersome:

• The first number in the bottom row of our synthetic division setup becomes the leading coefficient of the quotient polynomial.
• Follow the product and sum steps through synthetic division to fill all coefficients of the quotient.

The final expression \(-6x^2 - 8x - 12\) represents the simplified form of the original polynomial once it's been divided by the divisor, capturing all factors except the remainder.

Remainder Theorem

The remainder theorem is a useful concept stating that if a polynomial \(f(x)\) is divided by a linear polynomial \((x-c)\), the remainder of this division is the same as \(f(c)\). In our exercise, the remainder was found to be \(-22\).

This means that if we substitute \(x = \frac{3}{2}\) into the original polynomial \(-6x^3 + x^2 - 4\), it will equal the remainder, confirming the result of our synthetic division.

• The remainder appears as the last value in the bottom row of the synthetic division setup.
• Including the remainder in the final expression ensures the division accounts for all parts of the polynomials.

The concept also aids in understanding if full division occurred or if any of the original polynomial is left unaccounted for, represented by \(-\frac{22}{2x-3}\) in the solution.