Question

Find the quotient and the remainder. Check your work by verifying that Quotient \(\cdot\) Divisor \(+\) Remainder \(=\) Dividend $$ -4 x^{3}+x^{2}-4 \text { divided by } x-1 $$

Short answer

Quotient: \( -4x^2 - 3x - 3 \), Remainder: \( -7 \)

Step-by-step solution

- Setup Polynomial Division

Write down the dividend \( -4x^3 + x^2 - 4 \) and the divisor \( x - 1 \). Arrange it for polynomial long division as follows:\[ \frac{-4x^3 + x^2 + 0x - 4}{x - 1} \]Note that a 0 coefficient is added for the missing \( x \) term.

- Divide Leading Terms

Divide the leading term of the dividend \( -4x^3 \) by the leading term of the divisor \( x \):\[ \frac{-4x^3}{x} = -4x^2 \]This is the first term of the quotient.

- Multiply and Subtract

Multiply \( -4x^2 \) by the divisor \( x - 1 \):\[ -4x^2 \cdot x + (-4x^2 \cdot -1) = -4x^3 + 4x^2 \]Subtract this result from the original dividend:\[-4x^3 + x^2 - 4 - (-4x^3 + 4x^2) = -3x^2 - 4 \]

- Repeat Division

Divide \( -3x^2 \) by \( x \):\[ \frac{-3x^2}{x} = -3x \]This is the next term of the quotient.

- Multiply and Subtract Again

Multiply \( -3x \) by the divisor \( x - 1 \):\[ -3x \cdot x + (-3x \cdot -1) = -3x^2 + 3x \]Subtract this from the previous remainder:\[ -3x^2 - 4 - (-3x^2 + 3x) = -3x - 4 \]

- Final Division and Remainder

Divide \( -3x \) by \( x \):\[ \frac{-3x}{x} = -3 \]This is the next term of the quotient.Multiply \( -3 \) by the divisor \( x - 1 \):\[ -3 \cdot x + (-3 \cdot -1) = -3x + 3 \]Subtract this from the previous remainder:\[ -3x - 4 - (-3x + 3) = -7 \]So, the remainder is \( -7 \).

- Verify Answer

To verify, check if Quotient \( \cdot Divisor + Remainder = Dividend \):Quotient: \( -4x^2 - 3x - 3 \)Divisor: \( x - 1 \)Remainder: \( -7 \)Calculate:\[ (-4x^2 - 3x - 3)(x - 1) + (-7) = -4x^3 + x^2 - 4 \]This is equal to the original dividend, verifying the result.

Key concepts

Understanding Long Division with Polynomials

Polynomial long division works similarly to numerical long division but involves variables.
This method helps to determine how many times a polynomial (the divisor) can fit into another polynomial (the dividend).
Here's how it works step-by-step:

• Set up the long division by writing the dividend under the long division symbol and the divisor outside.
• Match the highest degree term in the dividend with the highest degree term in the divisor.
• Write the resulting quotient term directly above its corresponding term in the dividend.
• Multiply the quotient term by the entire divisor, then subtract the result from the current dividend.
• Repeat the steps until you can no longer divide or until you've worked down to a term of a lower degree than the divisor.

Every step involves simplifying and reducing the polynomial just like numerical long division. This process gives you the quotient and potentially a remainder.

The Role of Remainders in Polynomial Division

In polynomial division, the remainder is what’s left after dividing the polynomials completely until the degree of the remaining term is less than the degree of the divisor.
If there’s no remainder, the divisor perfectly divides the dividend.
For instance, when we divided -4x^3 + x^2 - 4 by x - 1, we were left with a remainder of -7.
To find the remainder, simply follow these steps:

• Perform division until the degree of the remainder is less than that of the divisor.
• Whatever is left over is the remainder.

This remainder tells us what’s left over, confirming that the polynomial didn’t divide evenly.

Quotients in Polynomial Division

The quotient is the result of the division process, where we divide the dividend polynomial by the divisor polynomial.
It consists of all the terms obtained during the long division process.
For the example given - -4x^3 + x^2 - 4 divided by x - 1, the quotient is -4x^2 - 3x - 3.
Finding the quotient involves:

• Dividing the highest degree terms of the dividend by the highest degree term of the divisor.
• Subtracting the result after multiplying back to reduce the dividend step by step.

Each term in the quotient helps to zero in on getting the correct remainder.

Verification of Polynomial Division Results

Verification is the process to ensure that the polynomial division has been done accurately.
To verify the result, multiply the quotient by the divisor and add the remainder.
Here’s our verification for the given problem:
Quotient: -4x^2 - 3x - 3
Divisor: x - 1
Remainder: -7
Calculation: ( -4x^2 - 3x - 3)(x - 1) + (-7) = -4x^3 + x^2 - 4
Since this result matches the original dividend ( -4x^3 + x^2 - 4), we can confirm our division is correct.
Verification helps confirm the accuracy of the division and ensures that no mistakes were made in the process.

Understanding Polynomials in Division

Polynomials are algebraic expressions that can consist of variables, coefficients, and exponents.
They can take various forms like: < br>

• Monomials (e.g., 3x, -5, 2x^2)
• Binomials (e.g., x + 1, 4x^2 - 9)
• Trinomials (e.g., x^2 + 3x + 2)

In polynomial division, understanding and arranging these components properly determines the success of the division process.
Notations and terms like degrees and orders help in aligning the terms properly for division.
The higher-degree (largest exponent) terms are always handled first. This requires a good grasp of polynomial structure to simplify and complete the long division accurately.