Problem 22 - College Algebra with Modeling and Visualization

Question

Divide Check your answer. $$\frac{12 x^{3}-14 x^{2}+7 x-7}{3 x-2}$$

Step-by-Step Solution

Answer

The quotient is \(4x^2 - 2x + 1\) with a remainder of \(-5\).

Step 1: Setup and Start Division

Set up the long division by dividing the polynomial \(12x^3 - 14x^2 + 7x - 7\) by \(3x - 2\). First, focus on the leading term \(12x^3\) and divide it by the leading term of the divisor \(3x\), which gives \(4x^2\). This is the first term of the quotient.

Step 2: Multiply and Subtract

Multiply \(4x^2\) by the entire divisor \(3x - 2\) to get \(12x^3 - 8x^2\). Subtract this result from the original polynomial \(12x^3 - 14x^2 + 7x - 7\), which gives the new polynomial: \(-6x^2 + 7x - 7\).

Step 3: Repeat for Next Term

Divide the new leading term \(-6x^2\) by the divisor's leading term \(3x\), resulting in \(-2x\). Multiply \(-2x\) by \(3x - 2\) to obtain \(-6x^2 + 4x\). Subtract \(-6x^2 + 4x\) from \(-6x^2 + 7x - 7\), leaving a new polynomial: \(3x - 7\).

Step 4: Final Term in Division

Divide the new leading term \(3x\) by \(3x\), which results in \(1\). Multiply \(1\) by the divisor \(3x - 2\) to get \(3x - 2\). Subtract \(3x - 2\) from \(3x - 7\), giving the remainder \(-5\).

Step 5: Conclusion and Verification

The division resulted in the quotient \(4x^2 - 2x + 1\) and a remainder of \(-5\). To verify, reconstruct the original expression by multiplying the quotient \(4x^2 - 2x + 1\) by the divisor \(3x - 2\) and add the remainder \(-5\). Ensure the result is \(12x^3 - 14x^2 + 7x - 7\), confirming the division is correct.

Key Concepts

Long Division

Polynomial long division is a process much like the long division we use with numbers. The procedure is helpful when dividing a polynomial by another polynomial, specially when the divisor is of a lower degree.
The long division method involves the following steps:

• Starting by dividing the leading term of the dividend by the leading term of the divisor. In this case, it's dividing the leading term of our dividend, which is \(12x^3\), by the leading term of our divisor, \(3x\), resulting in \(4x^2\).
• Next, multiply this result by the entire divisor, here it results in \(12x^3 - 8x^2\).
• Subtract the multiplication result from the original polynomial, and bring down the next term if needed. This subtraction gives us a new polynomial of \(-6x^2 + 7x - 7\).

We continue this process with the new polynomial, repeating similar steps until we cannot divide the leading term anymore. This results in a quotient and possibly a remainder.

Quotient and Remainder

The result from polynomial division is expressed as a quotient and remainder.
In our example, after using long division, we found the quotient \(4x^2 - 2x + 1\) and a remainder of \(-5\).

• The quotient is what you get from applying the division steps covered previously.
• The remainder is what's left once you cannot divide anymore, in this case, after dealing with lower degree terms compared to the divisor.

A division of the form \(\frac{P(x)}{Q(x)} = Q(x) + \frac{R(x)}{Q(x)}\) helps us verify our processing. If the remainder is zero, \(Q(x)\) is a factor of \(P(x)\). In most polynomial division problems, obtaining a numerical or polynomial remainder, like \(-5\) here, is common.

Verification of Division

Verification ensures that every step of the division was accurate. This is crucial to confirm the quotient and remainder are correct.
To verify, multiply the quotient \(4x^2 - 2x + 1\) by the divisor \(3x - 2\) and add the remainder \(-5\). This computation should bring us back to the original polynomial.
Performing these operations:

• Multiplying \(4x^2 - 2x + 1\) and \(3x - 2\), results in \(12x^3 - 14x^2 + 7x - 2\).
• Adding the remainder \(-5\) gives you \(12x^3 - 14x^2 + 7x - 7\).

This demonstrates our division was executed accurately, as the reconstructed expression matches the initial polynomial. Verification is an essential step in ensuring consistency and accuracy in mathematical operations.