Question

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

Short answer

Quotient: \(4x - 3\), Remainder: \(x + 1\)

Step-by-step solution

Title - Setup the Division

Setup the polynomial division of \[4x^3 - 3x^2 + x + 1\] by \[x^2\]. Write in long division format:

Title - Divide the Leading Terms

Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(x^2\):\[4x^3 \big/ x^2 = 4x\]

Title - Multiply and Subtract

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

Title - Repeat the Division

Now divide the new leading term (\(-3x^2\)) by the leading term of the divisor (\(x^2\)):\[-3x^2 \big/ x^2 = -3\]

Title - Multiply and Subtract Again

Multiply \(-3\) by the divisor \(x^2\):\[-3 \times x^2 = -3x^2\]Subtract this from the result of the previous subtraction:\[-3x^2 + x + 1 - (-3x^2) = x + 1\]

Title - Identify Quotient and Remainder

The quotient is the sum of the results from Steps 2 and 4:\[4x - 3\]and the remainder is the term that can't be further divided:\[x + 1\]

Title - Verify the Result

Verify by multiplying the quotient by the divisor and adding the remainder:\[(4x - 3) \times x^2 + (x + 1) = 4x^3 - 3x^2 + x + 1\]This matches the original dividend.

Key concepts

Polynomial Division

Polynomial division helps simplify complex expressions by dividing one polynomial by another. The main goal is to find how many times the divisor fits into the dividend. This process is similar to dividing numbers but involves variables. In our exercise, we'll be dividing \(4x^3 - 3x^2 + x + 1\) by \(x^2\). Polynomial division can be performed using long division or synthetic division methods.

Long division is particularly useful when the divisor or dividend has more than one term. It involves a series of steps to systematically reduce the dividend until no further division is possible. Each step consists of dividing, multiplying, and subtracting, just as you would in numerical long division.

Quotient and Remainder

In polynomial division, the end result includes a quotient and a remainder. The quotient is the polynomial you get after dividing the dividend by the divisor, whereas the remainder is what's left over and cannot be divided by the divisor.

• In our given problem: The quotient after dividing \(4x^3 - 3x^2 + x + 1\) by \(x^2\) is \(4x - 3\).
• The remainder, which cannot be simplified further by \(x^2\), is \(x + 1\).

These two parts are crucial because they fully describe the result of the division and can further help in simplifying and solving polynomial equations.

Verification in Algebra

Verification is a key step to ensure that our division is correct. We can verify our solution by multiplying the quotient by the divisor and then adding the remainder. The result should match the original polynomial (dividend). For our problem:

When you multiply the quotient \(4x - 3\) by the divisor \(x^2\) and then add the remainder \(x + 1\), you get: \( (4x - 3) \times x^2 + (x + 1) = 4x^3 - 3x^2 + x + 1 \). This matches the original polynomial, confirming that our division is correct.

This step is essential for avoiding mistakes and ensuring the integrity of your solution.

Long Division Method

The long division method breaks down the division process into manageable steps. Let's summarize the steps we've used:

• Write the polynomials in long division format: Dividend \(4x^3 - 3x^2 + x + 1\) and divisor \(x^2\).
• Divide the leading term of the dividend by the leading term of the divisor: \( 4x^3 \big/ x^2 = 4x \).
• Multiply the quotient \(4x\) by the divisor \(x^2\) and subtract the result from the original polynomial.
• Bring down the next term and repeat the steps with the new polynomial: \( -3x^2 \big/ x^2 = -3 \) and subtract again.
• Continue until no further division is possible. Identify the final quotient and remainder.

These steps illustrate how you can turn a potentially complicated process into a series of simple operations, making polynomial division more accessible and easier to understand.