Problem 2 - Precalculus: Functions and Graphs

Question

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=3 x^{4}+2 x^{3}-x^{2}-x-6 ; \quad p(x)=x^{2}+1$$

Step-by-Step Solution

Answer

Quotient: \(3x^2 + 2x - 4\), Remainder: \(-3x - 2\).

Step 1: Setup the Division

We will perform polynomial long division to find the quotient and remainder. Given the polynomials \(f(x) = 3x^4 + 2x^3 - x^2 - x - 6\) and \(p(x) = x^2 + 1\), set up the division in a long division format with \(f(x)\) as the dividend and \(p(x)\) as the divisor.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(3x^4\) by the leading term of the divisor \(x^2\). This gives \(3x^2\), which will be the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(3x^2\) by \(x^2 + 1\) to get \(3x^4 + 3x^2\). Subtract \(3x^4 + 3x^2\) from the original polynomial \(3x^4 + 2x^3 - x^2 - x - 6\) to get the new dividend \(2x^3 - 4x^2 - x - 6\).

Step 4: Repeat Division for New Dividend

Divide the leading term of the current dividend \(2x^3\) by the leading term of the divisor \(x^2\), which gives \(2x\). Add \(2x\) to the quotient.

Step 5: Multiply and Subtract Again

Multiply \(2x\) by \(x^2 + 1\) to get \(2x^3 + 2x\). Subtract this from \(2x^3 - 4x^2 - x - 6\) to get the new dividend \(-4x^2 - 3x - 6\).

Step 6: Final Division Step

Divide the leading term of the new dividend \(-4x^2\) by \(x^2\), which gives \(-4\). Add \(-4\) to the quotient.

Step 7: Final Multiplication and Subtraction

Multiply \(-4\) by \(x^2 + 1\) to get \(-4x^2 - 4\). Subtract this from \(-4x^2 - 3x - 6\) to get the remainder \(-3x - 2\).

Step 8: Write the Result

The final quotient is \(3x^2 + 2x - 4\) and the remainder is \(-3x - 2\). Therefore, the division of \(f(x)\) by \(p(x)\) gives:\[f(x) = (x^2 + 1)(3x^2 + 2x - 4) + (-3x - 2)\]

Key Concepts

Quotient

In polynomial division, the quotient is the result you get when you divide one polynomial by another. It represents how many times the divisor goes into the dividend completely. When solving the exercise of dividing the polynomial \(f(x) = 3x^4 + 2x^3 - x^2 - x - 6\) by \(p(x) = x^2 + 1\), we perform long division similar to arithmetic division.
The quotient obtained from this process is \(3x^2 + 2x - 4\).
This means that \(x^2 + 1\) can fit into \(3x^4 + 2x^3 - x^2 - x - 6\) this many times, with some remainder left over.
By calculating the quotient, we see the polynomial order decrease as each term accurately reflects the powers in the original polynomial. This provides an efficient method to express the original dividend in terms of the divisor and the found quotient.

Remainder

In polynomial division, the remainder is what's left over after dividing the dividend by the divisor. It's similar to the leftover amount you might have when dividing whole numbers. In our polynomial division case, after using long division, the remainder we find is \(-3x - 2\).
This remainder is a polynomial of lower degree than the divisor \(x^2 + 1\).
Knowing the remainder is crucial because it tells us that the division didn’t perfectly resolve itself into the divisor, which is common when dividing polynomials.
The complete expression for this problem is therefore:

• \( f(x) = (x^2 + 1)(3x^2 + 2x - 4) + (-3x - 2) \)

This equation confirms that the original polynomial is composed of our divisor multiplied by the quotient, plus this remainder.

Polynomial Long Division

Polynomial long division is a systematic method for dividing polynomials similar to long division with numbers. This method is used to find both the quotient and the remainder when dividing one polynomial by another.
1. **Setup:** Align the dividend and divisor similarly to numbers, with the dividend inside the division symbol.
2. **Divide Leading Terms:** Start by dividing the leading term of the dividend by the leading term of the divisor.
3. **Multiply and Subtract:** Multiply the resulting quotient term by the divisor and subtract it from the dividend.
4. **Repeat:** Continue this process with the new dividend formed after subtraction until the degree of the new dividend is less than that of the divisor.
This method ensures a fair understanding of polynomial relationships and handling large expressions, building conceptual understanding needed for algebraic manipulation.

Leading Term Division

Leading term division is the first step in polynomial long division. This is where the division process focuses initially on the highest degree terms of the dividend and divisor. It's key because it simplifies complex polynomial expressions into workable chunks.
The steps are pretty straightforward:

• Identify the leading term of both the dividend and divisor.
• Divide the leading term of the dividend by the leading term of the divisor.
• This gives the first term of the quotient.

For our exercise, this sequence began with dividing \(3x^4\) by \(x^2\), resulting in \(3x^2\), the starting term of the quotient.
Mastering leading term division is essential as it guides the rest of the long division process by gradually focusing on and simplifying the remaining parts of the polynomial.