Problem 15 - College Algebra with Modeling and Visualization

Question

Divide Check your answer. $$\frac{x^{4}-3 x^{3}-x+3}{x-3}$$

Step-by-Step Solution

Answer

The quotient is \(x^{3} - 1\) with a remainder of 0.

Step 1: Set up the Division

We will use polynomial long division to divide \(x^{4} - 3x^{3} - x + 3\) by \(x - 3\). First, set up the division with \(x^{4} - 3x^{3} - x + 3\) as the dividend and \(x-3\) as the divisor.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend, \(x^{4}\), by the leading term of the divisor, \(x\), to get \(x^{3}\). Write \(x^{3}\) above the division line.

Step 3: Multiply and Subtract

Multiply \(x^{3}\) by \(x - 3\), which gives \(x^{4} - 3x^{3}\). Subtract this from the original dividend to get the new dividend \(0x^{4} + 0x^{3} - x + 3\), simplifying to \(-x + 3\).

Step 4: Repeat the Division

Now divide the first term of the new dividend, \(-x\), by \(x\) to get \(-1\). Write \(-1\) above the division line, next to \(x^{3}\).

Step 5: Multiply and Subtract Again

Multiply \(-1\) by \(x - 3\) to get \(-x + 3\). Subtract this from \(-x + 3\), which results in a remainder of 0.

Step 6: Verify the Result

After completing the division, our quotient is \(x^{3} - 1\) with a remainder of 0. We verify this by multiplying the quotient \(x^{3} - 1\) by the divisor \(x - 3\) to ensure it equals the original dividend \(x^{4} - 3x^{3} - x + 3\). This confirms that the division is correct.

Key Concepts

Dividing Polynomials

Polynomial long division is similar to numerical long division, but instead of numbers, we deal with variables and powers. The main goal is to simplify the polynomial by dividing it into parts. First, identify the dividend (the polynomial you're dividing) and the divisor (the polynomial you are dividing by). For example, dividing \(x^{4} - 3x^{3} - x + 3\) by \(x - 3\) sets our dividend as \(x^{4} - 3x^{3} - x + 3\) and the divisor as \(x - 3\).

Here’s how it works:

• Align the polynomials as you would in numerical long division, making sure each term is in descending order of powers.
• Divide the highest degree term in the dividend by the highest degree term in the divisor. This gives the first term of the quotient.
• Multiply the entire divisor by this first term of the quotient and subtract the result from the original dividend.
• Repeat the steps with the new dividend obtained after subtraction until all terms are accounted for.
• The final expression above the division line gives the quotient, and any remainder will be beneath it.

Dividing polynomials is a critical skill in algebra, as it simplifies complex problems into more manageable parts.

Algebraic Techniques

Algebraic techniques are essential when dealing with polynomial division. They include strategic operations such as multiplication, subtraction, and the use of variables and their powers. Mastering these techniques allows you to manipulate and solve expressions effectively.

In our example, after dividing \(x^{4}\) by \(x\) to get \(x^{3}\), the next step involves multiplication and subtraction:

• Multiply \(x^{3}\) by the divisor \(x - 3\) to yield \(x^{4} - 3x^{3}\).
• Subtract this result from \(x^{4} - 3x^{3} - x + 3\), which simplifies the problem to a new dividend, \(-x + 3\).
• This process is repeated with the new dividend's first term, in this case, \(-x\).

These algebraic manipulations refine the polynomial until only a remainder is left, ensuring solutions are both accurate and efficient.

Remainder Verification

Remainder verification is an important step in polynomial division to check the accuracy of your problem-solving process. Once you've completed your division, how do you know your work is correct? This is where remainder verification comes in.

For the division of \(x^{4} - 3x^{3} - x + 3\) by \(x - 3\), we reached a quotient of \(x^{3} - 1\) with a remainder of 0. To verify:

• Multiply the quotient \(x^{3} - 1\) by the divisor \(x - 3\).
• Calculate to see if this product equals the original dividend \(x^{4} - 3x^{3} - x + 3\).

If the product matches the original dividend, the division is verified as correct. Remainder verification ensures confidence in your calculations and understanding of the process.