Question

Use long division to divide. Divisor \(x=1\) Dividend $$3 x^{2}-7 x+4$$

Short answer

The quotient of division is \(3x - 4\) and the remainder is 0.

Step-by-step solution

Divide the first term

The first term of the dividend is \(3x^2\). Divide this term by the divisor, \(x - 1\). This gives \(3x\). This is the first term of the quotient.

Multiply the divisor by the quotient's first term

Next, multiply the divisor, \(x - 1\), by the quotient's first term, \(3x\). This gives \(3x^2 - 3x\).

Subtract the product from the first two terms of the dividend

Subtract the product from the first two terms of the dividend. \(3x^2 -7x - (3x^2 - 3x) = -4x\). This is the new dividend.

Divide the new dividend

Now, divide the new dividend, \(-4x\), by the divisor, \(x - 1\). This gives -4, which is the second term of the quotient.

Multiply the divisor by the quotient's second term

Then, multiply the divisor, \(x - 1\), by the quotient's second term, -4. This gives \(-4x + 4\).

Subtract the product from the new dividend and the last term of original dividend

Subtract the product from the new dividend and the last term of the original dividend. This gives the remainder. \( -4x + 4 - (-4x + 4) = 0 \). The remainder is 0.

Key concepts

Long Division

Long division is a method used to divide polynomials, and it's similar to the long division technique used with numbers. The process involves several steps that allow you to find the quotient and remainder when dividing one polynomial by another.
It helps you break down complex division into smaller, manageable parts.
This involves dividing the terms one at a time and systematically subtracting to find the next part of the quotient until either a remainder is left or the division is exact (remainder of zero).
When dealing with polynomials, such as in this example, you align terms based on their degree, ensuring accurate subtraction at each step. Using long division helps illustrate the relationship between the divisor, dividend, and the resulting quotient and remainder.

Divisor

The divisor in polynomial long division is the polynomial you are using to divide into the dividend. In our example, the divisor is given as \(x - 1\).
The divisor plays a crucial role in determining how many times it goes into the dividend, thus directly affecting the quotient.

• To start, compare the leading term of the divisor to the leading term of the dividend.
• Then, multiply the entire divisor by the appropriate term in the quotient.
• This also impacts the subtraction step, which moves the process forward.

If the divisor was different, the entire division process would change, resulting in a different quotient and possibly a different remainder.

Dividend

In a division problem, the dividend is the polynomial that you are dividing. In this example, the dividend is \(3x^2 - 7x + 4\).
The role of the dividend is to be broken down through the division process, becoming smaller at each step as you find parts of the quotient.

• Each term of the dividend is important because it dictates the number of steps needed.
• Starting with the highest degree term, you systematically divide by the divisor and subtract to find the next dividend in the sequence.

This process continues until you reach the constant term or a point where the remainder is zero.

Quotient

The quotient is the result of the division, representing how many times the divisor fits into the dividend. As you proceed with polynomial division, the quotient is built up term by term.

• First, you divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient, which in this case was \(3x\).
• Then, you continue this process with the new resulting dividend until all terms are accounted for, yielding the whole quotient.

In the presented example, the quotient is \(3x - 4\), which means \(x - 1\) goes into \(3x^2 - 7x + 4\) evenly without any remainder at the end.

Remainder

After dividing the entire dividend by the divisor, you may be left with a remainder. This is what remains once the division process is completed.
For the example provided, the remainder is \(0\), indicating that \(x - 1\) divides perfectly into the dividend without leaving any excess.

• The remainder is found at the final step after subtracting the last set of terms.
• If the remainder is not zero, the division would not be exact, and it would need to be expressed as a fraction over the divisor.

Understanding the remainder is crucial, as it can indicate factors of the original polynomial and plays a role in polynomial division theorem.