Problem 4 - College Algebra

Question

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \left(x^{3}-2 x^{2}-5 x+6\right) \div(x-3) $$

Step-by-Step Solution

Answer

The quotient \(q(x) = x^{2}+2x-1\) and the remainder \(r(x) = 9\).

Step 1: Dividing leading terms

Start by dividing the leading term of the numerator, \(x^{3}\), by the leading term of the divisor, \(x\). This gives \(x^{2}\). Write this term above the line.

Step 2: Multiplying and subtracting

Next, multiply \(x^{2}\) by the divisor, \(x-3\), which gives \(x^{3}-3x^{2}\). Write this result under the original numerator, then subtract to find the next term. This gives \(2x^{2}-5x+6\).

Step 3: Repeat division

Repeat the process with the new term: divide the leading term, \(2x^{2}\), by \(x\) to get \(2x\). Write this term above the line, next to \(x^{2}\).

Step 4: Multiplying and subtracting

Multiply \(2x\) by the divisor, \(x-3\), to get \(2x^{2}-6x\). Write this under the last remainder and subtract to get \(-x+6\).

Step 5: Final division step

Carry out the division one more time with the new remainder: divide \(-x\) by \(x\) to get \(-1\). Write this above the line, next to \(2x\).

Step 6: Getting the remainder

After subtraction, the remainder \(r(x) = 9\). This is a term of lesser degree than that of the divisor, so the division process is complete.

Key Concepts

Quotient and Remainder

When dividing polynomials using long division, much like how it is done with numbers, we find a quotient and a remainder. The quotient is the result of the division, and the remainder is what is left after dividing.
In the case of polynomials, the remainder must have a smaller degree than the divisor, which indicates the end of the division process.

• The quotient, \( q(x) \), represents the polynomial result of the division process.
• The remainder \( r(x) \) is what remains. It's the part left over that cannot be divided further by the divisor.

Consider our exercise: Dividing \( (x^3 - 2x^2 - 5x + 6) \) by \( (x - 3) \), the long division gives us a quotient of \( x^2 + 2x - 1 \) and a remainder of \( 9 \).
This means:

• \( q(x) = x^2 + 2x - 1 \)
• \( r(x) = 9 \)

The quotient is what's written above the long division bar. It's like collecting each term from the division process, while the remainder is the leftover after the final subtraction.

Degree of Polynomials

The degree of a polynomial is determined by the highest power (or exponent) of the variable in the polynomial. It's a crucial concept when determining how to start the division process in polynomial long division and to verify when to stop.

• In the polynomial \( x^3 - 2x^2 - 5x + 6 \), the degree is 3, as the leading term is \( x^3 \).
• For the divisor \( x - 3 \), the degree is 1 since its leading term is \( x \).

Knowing the degree helps to ensure that the division proceeds correctly and tells us when we reach the end. When the remainder's degree is less than the divisor's degree, the process is complete.
To recap the division:

• The degree of the remainder \( r(x) \) should be less than the degree of the divisor, which was achieved with our remainder having degree 0 (as \( 9 \) is a constant).

Understanding polynomial degrees simplifies knowing how far to continue dividing and ensures clarity during polynomial operations.

Leading Term

The leading term in a polynomial is the term with the highest degree, and it guides the initial steps in polynomial long division. Recognizing the leading term and using it correctly is key to dividing effectively.

• For the polynomial \( x^3 - 2x^2 - 5x + 6 \), the leading term is \( x^3 \).
• In the divisor \( x - 3 \), the leading term is simply \( x \).

The very first step in long division involves the leading term of both the dividend (the polynomial being divided) and the divisor. By dividing the leading term of the dividend by that of the divisor, you find the first term of the quotient. This forms the backbone of what follows:

• Initially divide \( x^3 \) by \( x \) to get \( x^2 \), the first entry in the quotient.
• Each further step repeats a similar process with the leading term of the evolving remainder.

Starting from the leading term means you systematically reduce the degree of the polynomial, which is essential for accurate division results. It ensures that each step properly reduces the polynomial until you reach a remainder with a smaller degree than the divisor.