Problem 8 - College Algebra

Question

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \left(4 x^{2}-8 x+6\right) \div(2 x-1) $$

Step-by-Step Solution

Answer

The quotient \(q(x)\) is \(2x - 3\) and the remainder \(r(x)\) is \(3\).

Step 1: Divide

Divide the leading term in the dividend, or the term with the highest power of \(x\) (which is \(4x^2\)), by the leading term in the divisor (which is \(2x\)). This gives us our first term of the quotient, \(2x\).

Step 2: Multiply and Subtract

Multiply the divisor, \(2x -1\), by the first term of the quotient, \(2x\), obtained in Step 1 to get \(4x^2 - 2x\). Next, subtract this from the original dividend \(4x^2 - 8x + 6\) which gives us a new polynomial -\(6x + 6\).

Step 3: Repeat the Process

Repeat the process of Step 1 and Step 2 with this new polynomial - but this time, \(2x\) goes into \(-6x\) as \(-3\), giving us our next quotient term. Multiply the divisor, \(2x -1\), by \(-3\) to get \(-6x + 3\). Subtract this from \(-6x + 6\), the result from Step 2. The result is \(3\).

Step 4: Stop When Remainder's Degree is Less than the Divisor's

Stop this process once the degree of the remainder is less than the degree of the original divisor. Since \(3\) doesn't even have \(x\), we stop here.

Step 5: Conclusion

The polynomial division is done, the term you end up with as a result of this process is the remainder, which in our case is \(3\), while the terms we obtained in the process as the results of the division make up the quotient, which in our case are \(2x\) and \(-3\), so together our quotient is \(2x - 3\).

Key Concepts

Dividing Polynomials

Polynomial long division is a process very similar to long division of numbers. When dividing polynomials, you arrange the terms in descending order of powers of the variable, just as you would place numbers in descending place values for numerical division.

Here's a simplified breakdown:

• Start by dividing the highest degree term of the polynomial (the dividend) by the highest degree term of the divisor.
• Place the result of this division (the first term of the quotient) above the dividend.
• Multiply this term by the divisor and subtract the result from the dividend to find the remainder.
• Repeat the process, now using the new remainder as the dividend, until the degree of the remainder is less than the degree of the divisor.
• Whatever terms you've placed in the quotient area during this process make up your final quotient, and the last remainder is the remainder of the polynomial division.

In the given exercise, the dividend is \(4x^2 - 8x + 6\) and the divisor is \(2x - 1\). Following the steps mentioned leads to a quotient of \(2x - 3\) and a remainder of \(3\).

Synthetic Division

Synthetic division is an alternative, and often simpler, method to dividing polynomials as compared to long division—especially when dividing by a linear factor (a polynomial of the first degree).

The synthetic division process is as follows:

• Write down the coefficients of the dividend polynomial.
• Place the zero of the divisor (the solution of the divisor set equal to zero) to the left.
• Perform a series of operations involving these coefficients and the divisor's zero to find the new coefficients of the quotient polynomial and possibly a remainder.

Synthetic division is neater and more straightforward than long division, particularly when working with higher degree polynomials. It isn't applicable when the divisor is not a linear polynomial with a leading coefficient of 1. If our example was to be solved through synthetic division, we would only work with the coefficients and the zero of the divisor \(2x - 1\), which is \(\frac{1}{2}\). It's an efficient technique that significantly reduces the complexity of the division process.

Remainder Theorem

The Remainder Theorem is a result in algebra which gives us a quick way to find the remainder when a polynomial is divided by a linear divisor of the form \(x - c\).

According to the theorem:

• To find the remainder, simply evaluate the polynomial at the value of \(c\).

This means that if you're dividing a polynomial \(P(x)\) by \(x - c\), instead of performing long division, you can just calculate \(P(c)\), where \(c\) is the zero of \(x - c\).

In our original exercise, if you used the Remainder Theorem to find the remainder when dividing \(4x^2 - 8x + 6\) by \(2x - 1\), you would evaluate the polynomial at \(\frac{1}{2}\), the zero of the divisor. You would discover the remainder is \(3\), quickly confirming the result obtained through long division.