Problem 14 - Algebra and Trigonometry

Question

Find the quotient and remainder using long division. \(\frac{x^{3}-x^{2}-2 x+6}{x-2}\)

Step-by-Step Solution

Answer

Quotient: \(x^2 + x - 2\), Remainder: \(2\).

Step 1: Setup the Division

Write down the polynomial you're dividing, \(x^3 - x^2 - 2x + 6\), and the divisor, \(x - 2\). Set it up as a long division problem, with \(x^3 - x^2 - 2x + 6\) inside the division symbol and \(x - 2\) outside.

Step 2: Divide the First Term

Divide the first term of the dividend, \(x^3\), by the first term of the divisor, \(x\). This gives \(x^2\). Write \(x^2\) above the division bar as part of the quotient.

Step 3: Multiply and Subtract

Multiply \(x^2\) by \(x - 2\), which results in \(x^3 - 2x^2\). Write this product below the dividend. Subtract \(x^3 - 2x^2\) from \(x^3 - x^2 - 2x + 6\), giving \(x^2 - 2x + 6\).

Step 4: Repeat the Process

Divide \(x^2\) by the first term of \(x - 2\), which is \(x\). This result is \(x\). Write \(+x\) in the quotient next to \(x^2\). Multiply \(x\) by \(x - 2\) to get \(x^2 - 2x\) and subtract it from \(x^2 - 2x + 6\), resulting in \(0 + 6\).

Step 5: Final Division Cycle

Divide the term \(-2x\) by \(x\), yielding a quotient of \(-2\). Multiply \(-2\) by \(x - 2\) to get \(-2x + 4\). Subtract \(-2x + 4\) from \(-2x + 6\), resulting in a remainder of \(2\).

Step 6: Check the Remainder

Verify that the remainder \(2\) is less than the degree of \(x - 2\), which is 1, confirming that the division process is complete.

Key Concepts

Quotient and Remainder

When performing polynomial long division, two key elements arise: the quotient and the remainder. These terms help us understand the result of dividing one polynomial by another.

• Quotient: The quotient is the result of the division process itself. In polynomial division, it represents the polynomial obtained when the dividend is divided by the divisor. For example, in dividing \(x^3 - x^2 - 2x + 6\) by \(x - 2\), the quotient is \(x^2 + x - 2\).
• Remainder: The remainder is what is left after the division has been completed. For polynomials, the remainder can be seen as what's left over when no further division is possible within the terms of the dividend. In our example, the remainder is 2.

Understanding the quotient and remainder is crucial for solving polynomial division problems. These terms not only give us a complete picture of the division but are also foundational in further algebraic operations such as synthetic division and factoring.
Completing a division with a remainder smaller than the divisor's highest degree confirms the accuracy of the division.

Polynomial Division

Polynomial division is a process similar to long division with numbers. It involves dividing one polynomial, termed the dividend, by another polynomial, known as the divisor. The objective is to find both the quotient and the remainder.
The process is systematic:

• Set Up: Align the dividend and divisor similar to number long division.
• Divide: Divide the first term of the dividend by the first term of the divisor to begin building the quotient.
• Multiply and Subtract: Multiply the divisor by the term obtained, subtract it from the dividend to get a new polynomial.
• Repeat: Continue this process with the resulting polynomial until the degree of the remainder is less than the degree of the divisor.

Understanding polynomial division is vital for solving various algebraic equations efficiently.
It helps to break down complex expressions into simpler components, creating opportunities for deeper insights into the properties of polynomials.

Degrees of Polynomials

When dealing with polynomials, the concept of degrees is of prime importance. The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial \(x^3 - x^2 - 2x + 6\), the degree is 3 because the highest power of \(x\) is 3.
Understanding the degree of polynomials has several implications:

• Determining Divisions: When dividing polynomials, the degree of the dividend and divisor indicate how many steps of division are possible.
• Identifying Remainders: A division process concludes correctly when the remaining polynomial, the remainder, is of a lower degree than the divisor.
• Predicting Solutions: The degree provides insight into the number of potential roots or zeroes a polynomial may have.

In polynomial division, always check degrees to confirm the remainder cannot be further divided by the divisor. This ensures the division is complete, accounting for all potential divisions.