Polynomial Division A Comprehensive Guide To Dividing $10x^4 - 14x^3 - 10x^2 + 6x - 10$ By $x^3 - 3x^2 + X - 2$

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Polynomial division can seem daunting at first, but by breaking it down into smaller, manageable steps, you can master this essential algebraic technique. In this comprehensive guide, we'll walk through the process of dividing the polynomial $10x^4 - 14x^3 - 10x^2 + 6x - 10$ by $x^3 - 3x^2 + x - 2$. We'll not only find the quotient and remainder, but also delve into the underlying concepts to ensure you understand each step.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in arithmetic, but instead of dividing numbers, we're dividing polynomials. The goal is to find two polynomials: the quotient (the result of the division) and the remainder (the part that's left over). The general form of polynomial division can be expressed as:

Dividend = (Divisor × Quotient) + Remainder

In our case:

• Dividend: $10x^4 - 14x^3 - 10x^2 + 6x - 10$ (The polynomial being divided)
• Divisor: $x^3 - 3x^2 + x - 2$ (The polynomial we are dividing by)
• Quotient: The polynomial we will find as the result of the division.
• Remainder: The polynomial left over after the division, which will have a degree less than the divisor.

Before we begin, it's crucial to ensure both the dividend and the divisor are written in descending order of powers of the variable (in this case, x). Both polynomials are already in the correct format, so we can proceed directly to the division process.

Step-by-Step Polynomial Division

Let's dive into the step-by-step process of dividing $10x^4 - 14x^3 - 10x^2 + 6x - 10$ by $x^3 - 3x^2 + x - 2$:

Step 1: Set up the Long Division

Write the problem in the long division format, similar to how you would set up long division with numbers:

                  __________________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10

Step 2: Divide the Leading Terms

Focus on the leading terms of both the divisor ($x^3$) and the dividend ($10x^4$). Ask yourself: "What do I need to multiply $x^3$ by to get $10x^4$?" The answer is $10x$. Write $10x$ above the line, aligning it with the $x$ term in the dividend:

                  10x______________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10

Step 3: Multiply the Quotient Term by the Divisor

Multiply the $10x$ (the first term of our quotient) by the entire divisor ($x^3 - 3x^2 + x - 2$):

$10x * (x^3 - 3x^2 + x - 2) = 10x^4 - 30x^3 + 10x^2 - 20x$

Write the result below the dividend, aligning like terms:

                  10x______________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                      10x^4 - 30x^3 + 10x^2 - 20x

Step 4: Subtract

Subtract the expression you just wrote from the corresponding terms in the dividend. Remember to distribute the negative sign:

$(10x^4 - 14x^3 - 10x^2 + 6x - 10) - (10x^4 - 30x^3 + 10x^2 - 20x) = 16x^3 - 20x^2 + 26x - 10$

Write the result below the line:

                  10x______________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                      10x^4 - 30x^3 + 10x^2 - 20x
                      ----------------------------------
                              16x^3 - 20x^2 + 26x - 10

Step 5: Bring Down the Next Term

There are no more terms to bring down in this case, as we've already included the constant term (-10) in our subtraction result.

Step 6: Repeat the Process

Now, treat the result of the subtraction ($16x^3 - 20x^2 + 26x - 10$) as the new dividend. Repeat steps 2-4:

• Divide the leading terms: What do you need to multiply $x^3$ by to get $16x^3$? The answer is $16$. Write $+16$ next to $10x$ in the quotient.

                  10x + 16__________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                      10x^4 - 30x^3 + 10x^2 - 20x
                      ----------------------------------
                              16x^3 - 20x^2 + 26x - 10

• Multiply the quotient term by the divisor: $16 * (x^3 - 3x^2 + x - 2) = 16x^3 - 48x^2 + 16x - 32$

Write the result below the new dividend, aligning like terms:

                  10x + 16__________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                      10x^4 - 30x^3 + 10x^2 - 20x
                      ----------------------------------
                              16x^3 - 20x^2 + 26x - 10
                              16x^3 - 48x^2 + 16x - 32

• Subtract: $(16x^3 - 20x^2 + 26x - 10) - (16x^3 - 48x^2 + 16x - 32) = 28x^2 + 10x + 22$

Write the result below the line:

                  10x + 16__________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                      10x^4 - 30x^3 + 10x^2 - 20x
                      ----------------------------------
                              16x^3 - 20x^2 + 26x - 10
                              16x^3 - 48x^2 + 16x - 32
                              ----------------------------------
                                      28x^2 + 10x + 22

Step 7: Determine the Remainder

We stop the division process when the degree of the remaining polynomial ($28x^2 + 10x + 22$) is less than the degree of the divisor ($x^3 - 3x^2 + x - 2$). In this case, the degree of the remainder (2) is less than the degree of the divisor (3), so we have reached the end of the division.

Results: Quotient and Remainder

From our long division, we have found:

• Quotient: $10x + 16$
• Remainder: $28x^2 + 10x + 22$

Therefore, we can write:

$10x^4 - 14x^3 - 10x^2 + 6x - 10 = (x^3 - 3x^2 + x - 2)(10x + 16) + (28x^2 + 10x + 22)$

Expressing the Answer

Based on the original question's format, we can express the quotient and remainder as follows:

The quotient is 10x + 16 The remainder is 28$x^2$ + 10x + 22

Key Takeaways and Further Practice

Mastering polynomial division is crucial for various algebraic manipulations, including factoring polynomials, solving equations, and simplifying rational expressions. Remember these key takeaways:

• Organize your work: Use the long division format to keep track of terms and prevent errors.
• Focus on the leading terms: Divide the leading term of the dividend by the leading term of the divisor.
• Distribute carefully: Remember to distribute when multiplying the quotient term by the divisor and when subtracting.
• Stop when the degree of the remainder is less than the degree of the divisor.

To solidify your understanding, try practicing with various polynomial division problems. Start with simpler examples and gradually increase the complexity. You can also explore online resources and textbooks for additional practice problems and explanations.

By following this step-by-step guide and practicing consistently, you'll become proficient in polynomial division and gain confidence in your algebraic abilities.