Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Use long division to divide. Divisor \(x-4\) Dividend $$2 x^{3}-8 x^{2}+3 x-9$$

Step-by-Step Solution

Answer

The quotient is \(2x^2 + 3\) with a remainder -12.

Step 1: Set up the long division

Start by drawing the long division symbol and write the dividend \(2x^3 - 8x^2 + 3x - 9\) inside it and the divisor \(x - 4\) outside.

Step 2: Divide the first term of the dividend by the first term of the divisor

Divide the first term of the dividend, \(2x^3\), by the first term of the divisor, x. The result is \(2x^2\). Write this value on top of the long division symbol.

Step 3: Multiply the divisor by the result obtained from Step 2

Multiply the divisor, \(x - 4\), by the result get from step 2, \(2x^2\). The result should be \(2x^3 - 8x^2\). Write this value under the first two terms of the dividend and draw a line under it.

Step 4: Subtract and bring down the next term

Subtract the results from step 3, \(2x^3 - 8x^2\), from the respective terms in the dividend, which results in a new dividend. Add the next term from the dividend, 3x, to the result to get a new polynomial that we're ready to divide by: \(3x - 9\).

Step 5: Repeat steps 2 to 4

Now again, divide the first term of the new dividend, \(3x\), by the first term of the divisor, \(x\). This gives 3. Multiply the divisor by 3 and subtract the results from the new dividend, resulting in -12. This is the remainder. When there are no more terms to bring down, we stop here.

Step 6: Write down the complete quotient and remainder

Write down the complete quotient which is the values obtained from step 2 and step 5 put together, which were \(2x^2 + 3\), and the remainder which we got as -12. This completes the polynomial long division.

Key Concepts

Polynomial Division Steps

Mastering polynomial division involves understanding the process step-by-step. It's similar to long division you learned in elementary school, but with variables involved. Here's an intuitive way to tackle it:

• First, write the dividend (the polynomial being divided) and the divisor (what you're dividing by) in the long division format.
• Divide the first term of the dividend by the first term of the divisor. Place this result above the division bar.
• Multiply the entire divisor by the result from the step above, writing it underneath the appropriate terms of the dividend.
• Subtract to find the new dividend. If there are remaining terms in the original dividend, bring the next one down.
• Repeat the process of dividing, multiplying, and subtracting until there are no terms left to bring down.
• The quotient is the combination of all your results atop the division bar, and any remainder gets written as a remainder term.

Remember to pay attention to subtracting negative terms, as these simple mistakes can throw off your quotient and remainder. Clear handwriting and orderly work also minimize errors.

Dividing Polynomials

Dividing polynomials can initially seem daunting, but with practice, it becomes second nature. The goal is to determine how many times the divisor fits into the dividend. In algebra,

• You always start with the leading terms (those with the highest exponent).
• Ensure your polynomials are written in descending exponential order, which aids in aligning like terms.
• If your dividend misses terms (like if there's no x-squared term), insert a placeholder (0x^2) to keep things in line.

Consistency in process is key. Each step builds on the previous one, so any small mistake early on will affect the rest of your division. After a few repetitions, you'll start to notice patterns that will make predicting the next step easier, speeding up your work.

Algebraic Long Division

Just like numerical long division, algebraic long division is all about breaking down seemingly complex division problems into simpler steps. When dividing polynomials, particularly:

• All terms need to be considered, even those that might not have a visible impact at first glance, such as the zero coefficients for non-present terms.
• Keep track of your negative and positive signs—algebraic division often stumbles on missed sign changes.
• Make sure to line up the terms correctly under the division bar so that you are always subtracting like terms.
• Remember that the process continues until no more terms from the dividend can be divided by the leading term of the divisor.

With algebraic long division, neatness, and systematic work helps you avoid errors. It's a fundamental skill that unlocks the ability to handle more complex algebraic expressions.

Polynomial Remainder Theorem

The Polynomial Remainder Theorem offers a brilliant insight into division. It states that when a polynomial, f(x), is divided by a linear divisor like (x - k), the remainder is f(k). This theorem can streamline your work:

• If you simply need the remainder, you can directly substitute the k-value into the polynomial and calculate f(k) instead of doing long division.
• The theorem is incredibly useful for verifying your results or for checking the factors of polynomials.
• It's also the basis for the Factor Theorem, which tells us that if f(k) equals zero, then (x - k) is a factor of the polynomial.

Understanding the Polynomial Remainder Theorem is very helpful in simplifying complex polynomial division problems and can also be a great time-saver for quick calculations. Applying this concept, along with practicing division steps, makes the task of dividing polynomials more manageable and less time-consuming.