Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}-3 x^{2}+1\right) \div(x-1) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 + x^2 - 2x - 2\) with a remainder of \(-1\).

Step 1: Write the Polynomial in Standard Form

Ensure that the dividend polynomial \(x^4 - 3x^2 + 1\) is written in standard form. In this case, the polynomial is already in the correct form, which includes all terms from the highest degree to the constant, even if their coefficients are zero. Thus, the polynomial is \(x^4 + 0x^3 - 3x^2 + 0x + 1\).

Step 2: Identify the Divisor and Coefficient

The divisor is \(x - 1\). Synthetic division requires the dividend to be divided by \(x - c\); here, \(c = 1\). There's no need to adjust since the linear term's coefficient in the divisor is already 1.

Step 3: Set Up Synthetic Division

Using the polynomial coefficients 1, 0, -3, 0, 1 from \(x^4 + 0x^3 - 3x^2 + 0x + 1\), set up the synthetic division chart. Use \(c = 1\) as the number on the left side.

Step 4: Perform Synthetic Division

1. Bring down the first coefficient (1) to the bottom row.2. Multiply \(c\) (which is 1) by the number just written below the line (1) and place the result under the next coefficient (0).3. Add the numbers in the second column: \(0+1 = 1\).4. Repeat the process: multiply the result (1) by \(c\) (1), write it under the next coefficient (-3), and add: \(-3+1 = -2\).5. Continue this method for the remaining coefficients: multiply the result (-2) by 1, place it under 0, add to get: \(0 - 2 = -2\).6. Multiply -2 by 1, place it under 1, add to get: \(1 - 2 = -1\).

Step 5: Write the Quotient

The numbers at the bottom row of the synthetic division chart are the coefficients of the quotient polynomial. The quotient is \(x^3 + x^2 - 2x - 2\) with a remainder of \(-1\). Therefore, the result of the division is \(x^3 + x^2 - 2x - 2 - \frac{1}{x-1}\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to how you would use long division with numbers. In this process, you have two polynomials: the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by). The goal is to find both the quotient and the remainder when you divide the dividend by the divisor.

In this exercise, we are dividing the polynomial \(x^4 - 3x^2 + 1\) by \(x - 1\). Polynomial division can be done through traditional long division or a faster method called synthetic division, which is used when the divisor is a linear polynomial of the form \(x - c\). This technique simplifies the division process and involves working with the coefficients of the polynomial directly.

Quotient and Remainder

When performing polynomial division, the quotient is the result of the division and is expressed as a polynomial. The remainder is what is left over after the division is complete. The remainder will always have a degree less than the degree of the divisor.

In the solution provided, after dividing \(x^4 - 3x^2 + 1\) by \(x-1\) using synthetic division, the quotient is found to be \(x^3 + x^2 - 2x - 2\). The remainder in this case is \(-1\). This means that you can express the original division as:

• Quotient: \(x^3 + x^2 - 2x - 2\)
• Remainder: \(-1\)

Thus, the full expression of the division is \(x^3 + x^2 - 2x - 2 - \frac{1}{x-1}\), combining the quotient and the division of the remainder by the divisor.

Polynomial Coefficients

The coefficients of a polynomial are the numbers in front of each term in the polynomial expression. These coefficients are integral when using synthetic division because they are the values you manipulate throughout the division process.

For the polynomial \(x^4 - 3x^2 + 1\) written in standard form as \(x^4 + 0x^3 - 3x^2 + 0x + 1\), the coefficients are:

• 1 for \(x^4\)
• 0 for \(x^3\) (since there's no \(x^3\) term in the original polynomial)
• -3 for \(x^2\)
• 0 for \(x\) (since there's no \(x\) term in the original polynomial)
• 1 as the constant term

These coefficients are lined up to set up the synthetic division chart, which then allows you to perform the calculations more easily.

Higher-Degree Polynomials

Higher-degree polynomials, like the \(x^4\) term in our example, can seem intimidating but can be broken down into simpler parts through division methods like synthetic division.

When working with higher-degree polynomials, ensure your polynomial is in standard form. This involves listing every term in the polynomial from highest to lowest degree, including terms with a zero coefficient. This setup is critical for performing synthetic division accurately. In our exercise, after writing \(x^4-3x^2+1\) in standard form, it becomes \(x^4 + 0x^3 - 3x^2 + 0x + 1\). This form allows us to efficiently handle the polynomial division process, even with the highest degree term.