Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{3}-2 x^{2}-9 ; \quad x-3$$

Step-by-Step Solution

Answer

The quotient is \(x^2 + x + 3\).

Step 1: Set Up Division

Write the polynomial division problem in long division format, setting up \(x^3 - 2x^2 + 0x - 9\) as the dividend and \(x - 3\) as the divisor.

Step 2: Divide Leading Terms

Divide the leading term of the dividend \(x^3\) by the leading term of the divisor \(x\), which results in \(x^2\). Write \(x^2\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(x^2\) by the entire divisor \(x - 3\), which gives \(x^3 - 3x^2\). Subtract this product from the dividend, which results in \(x^3 - 2x^2 - (x^3 - 3x^2) = x^2\).

Step 4: Repeat Division Process

Bring down the next term \(0x\) to get \(x^2 + 0x\). Divide the leading term \(x^2\) by \(x\) to get \(x\). Write \(x\) above the division bar next to \(x^2\).

Step 5: Multiply and Subtract Again

Multiply \(x\) by \(x - 3\), yielding \(x^2 - 3x\). Subtract this from the current dividend, \(x^2 + 0x - (x^2 - 3x) = 3x\).

Step 6: Continue Division

Bring down the last term \(-9\) to form \(3x - 9\). Divide \(3x\) by \(x\) to get \(3\), and write \(3\) above the division bar.

Step 7: Final Multiplication and Subtraction

Multiply \(3\) by \(x - 3\) to obtain \(3x - 9\). Subtract from the current expression \(3x - 9 - (3x - 9) = 0\). There is no remainder.

Step 8: Write the Final Answer

The quotient of \(P(x)\) divided by \(x - 3\) is the expression written above the division bar: \(x^2 + x + 3\).

Key Concepts

Long Division

Long division in polynomials resembles numeric long division. It involves dividing a complex polynomial by a simpler one. Here, we deal with dividing the polynomial \(P(x) = x^3 - 2x^2 - 9\) by the binomial \(x - 3\).

• Set up the division: Write the dividend, including any missing power terms (here, add \(0x\)).
• Begin with the leading terms: Divide the leading term of the dividend by the leading term of the divisor.
• Continue the process: Subtract, bring down the next term, and repeat.

By following these steps meticulously, you can break down complex polynomial expressions into manageable parts for division.

Binomials

A binomial is a polynomial with exactly two terms. In the given problem, \(x - 3\) is the binomial divisor. Understanding binomials is crucial as it forms the basic block of polynomial division.
In this context:

• The binomial \(x - 3\) is the divisor.
• Always examine the structure of the binomial to correctly apply the division operation.
• While dividing, focus on the leading term as it guides the initial step in the division process.

Recognizing and working with such binomials sharpens understanding and facilitates problem-solving in algebraic contexts.

Quotients

The quotient is the result of the division operation. In polynomial division, the quotient reflects the simplified expression obtained when one polynomial is divided by another.
In our example, dividing \(P(x) = x^3 - 2x^2 - 9\) by \(x - 3\) gives the quotient:

• Start with the leading term and work through each step meticulously.
• The quotient \(x^2 + x + 3\) is developed as each part is completed.
• Ensure there is no remainder or understand if one exists.

The final quotient summarizes the division results without any leftover or remainder.

Degree of Polynomials

The degree of a polynomial is determined by the highest power of the variable in the expression. It helps in understanding the behavior and complexity of the polynomial.
For the polynomial \(P(x) = x^3 - 2x^2 - 9\):

• The degree is 3, given by the term \(x^3\).
• When dividing by the binomial \(x - 3\), understanding degrees helps predict the degree of the quotient (here, \(x^2 + x + 3\) has a degree of 2).
• Degree guides both division alignment and outcome expectations.

Grasping the degree helps in planning division steps and verifying the correctness of results.