Problem 11 - College Algebra

Question

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(3 x^{2}-5 x+4\right) \div(3 x+1) $$

Step-by-Step Solution

Answer

Quotient: \(x - 2\); Remainder: 6.

Step 1: Set up the Long Division

Write the division in long division format: \[ \text{Dividend: } 3x^2 - 5x + 4 \quad \text{Divisor: } 3x + 1 \]. The division setup looks like this:\[ 3x + 1 \; | \; 3x^2 - 5x + 4 \].

Step 2: Divide the First Term

Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{3x^2}{3x} = x \]. This will be the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(x\) by the divisor \(3x + 1\) to get \(3x^2 + x\). Subtract this from the original polynomial:\[(3x^2 - 5x + 4) - (3x^2 + x) = -6x + 4 \]. Write the result below the line.

Step 4: Repeat the Division

Take the new polynomial \(-6x + 4\) and divide the leading term \(-6x\) by the leading term of the divisor \(3x\): \[ \frac{-6x}{3x} = -2 \]. This is the next term of the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-2\) by \(3x + 1\) to get \(-6x - 2\). Subtract this from \(-6x + 4\):\[ (-6x + 4) - (-6x - 2) = 6 \]. The remainder is 6.

Step 6: Write the Quotient and Remainder

Combine the terms to write the final solution. The quotient from our division is: \[ x - 2 \]. The remainder is 6, so the complete division statement is: \[ 3x^2 - 5x + 4 = (3x + 1)(x - 2) + 6 \].

Key Concepts

Polynomial Division

Polynomial division is like regular long division but instead of numbers, we use polynomials. It helps us divide a large polynomial (the dividend) by a smaller one (the divisor). This process is crucial for simplifying expressions in algebra and solving polynomial equations. Here's how it works:

• You set up the division much like numerical long division, writing the dividend under a division bar and the divisor to the left.
• The goal is to find how many times the divisor can "fit" into parts of the dividend, step by step, leading to a quotient and possibly a remainder.

Understanding polynomial division can be daunting at first. However, breaking it down into clear, systematic steps, as seen in our example, makes it manageable.

Quotient and Remainder

The result of dividing one polynomial by another using long division results in a quotient and a remainder, similar to dividing numbers. The quotient is the polynomial you get after the division, which is written above the division bar. The remainder is what is left over after dividing as much as possible.

• In our example, the quotient is found to be \(x - 2\).
• The remainder happens to be a constant, which is 6 here.

The complete division is expressed as: \(3x^2 - 5x + 4 = (3x + 1)(x - 2) + 6\). This notation shows how the original polynomial can be reconstructed, verifying our division.

Step-by-Step Division

Breaking down polynomial division into simple, manageable steps is essential for understanding. Let's recap the essential steps:

• Step 1: Set up by writing the dividend and divisor in long division format.
• Step 2: Divide the first term of the dividend by the first term of the divisor to begin forming the quotient.
• Step 3: Multiply and subtract to find the new polynomial to work with.
• Step 4: Divide again, repeating the process to expand the quotient further.
• Step 5: Continue multiplying and subtracting until you cannot divide any further.

In each replication of these steps, you're chiseling away at the dividend until you're left with the remainder. This procedure demonstrates the principles at work in polynomial division, facilitating a deeper understanding of algebraic manipulation.