Problem 27 - College Algebra with Modeling and Visualization

Question

Divide the expression. $$\frac{8 x^{3}+10 x^{2}-12 x-15}{2 x^{2}-3}$$

Step-by-Step Solution

Answer

The quotient is \(4x + 5\).

Step 1: Set Up Polynomial Long Division

To divide the polynomial \(8x^3 + 10x^2 - 12x - 15\) by \(2x^2 - 3\), we start by writing the dividend \((8x^3 + 10x^2 - 12x - 15)\) and the divisor \((2x^2 - 3)\) in long division format.

Step 2: Divide First Terms

Divide the first term of the dividend \(8x^3\) by the first term of the divisor \(2x^2\). This gives \(\frac{8x^3}{2x^2} = 4x\). The first term of our quotient is \(4x\).

Step 3: Multiply and Subtract

Multiply \(4x\) by the entire divisor \(2x^2 - 3\): \(4x \times (2x^2 - 3) = 8x^3 - 12x\). Subtract this from the original dividend: \[(8x^3 + 10x^2 - 12x - 15) - (8x^3 - 12x) = 10x^2 + 0x - 15\].

Step 4: Repeat Division with New Polynomial

Now, divide the first term of the new polynomial \(10x^2\) by \(2x^2\): \(\frac{10x^2}{2x^2} = 5\). Add this to the quotient to get \(4x + 5\).

Step 5: Multiply and Subtract Again

Multiply \(5\) by \(2x^2 - 3\): \(5 \times (2x^2 - 3) = 10x^2 - 15\). Subtract this from the new dividend: \[(10x^2 + 0x - 15) - (10x^2 - 15) = 0\]. This results in a remainder of 0.

Step 6: Finalize the Quotient

Since the remainder is zero after performing the divisions, the final quotient is \(4x + 5\). This is the solution to the original division problem.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to the long division used in regular arithmetic. It helps to simplify complex polynomial expressions and find the quotient, just like dividing numbers. Here's how it works:

• Place the dividend (the polynomial to be divided) under the long division bracket and the divisor (the polynomial you are dividing by) outside the bracket.
• Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend.
• Bring down the next term and repeat the process with the new polynomial.
• Continue until all terms of the dividend have been used.

This method systematically breaks down the division process, allowing us to compute the quotient and, if possible, a remainder.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to express and solve equations by using variables. In polynomial division, algebra is crucial:

• Variables, such as \(x\), represent unknown values and allow us to form expressions.
• Algebraic expressions, like polynomials, consist of terms that are added, subtracted, or multiplied together.
• Understanding coefficients and exponents is important—these are the numerical and variable parts in terms like \(8x^3\).
• Algebraic techniques like polynomial division help in simplifying expressions and solving equations.

With a strong foundation in algebra, you can navigate through polynomial problems effectively and grasp these concepts with clarity.

Quotient and Remainder

In every division problem, whether it's arithmetic or polynomial division, the objective is to find the quotient and remainder. The quotient is what you obtain from dividing one polynomial by another, while the remainder is what’s left over when the division doesn't evenly divide the dividend.

• The quotient is the result you get after using polynomial long division, representing how many times the divisor fits into the dividend.
• Sometimes, a division may have a remainder, a term that cannot be further divided by the divisor.
• In our original problem, since the remainder is zero, it indicates that the dividend is perfectly divisible by the divisor.
• The expression \(4x + 5\) is the quotient, showing the simplified form after the division of our polynomial expression \(8x^3 + 10x^2 - 12x - 15\) by \(2x^2 - 3\).

Understanding quotient and remainder is vital as it helps determine the result of polynomial division tasks, highlighting whether it divides cleanly or not.