Problem 6 - College Algebra

Question

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(4 x^{2}-10 x+6\right) \div(4 x+2) $$

Step-by-Step Solution

Answer

Quotient is \(x - 3\) and remainder is \(12\).

Step 1: Set up the Division

Write the dividend, \(4x^2 -10x + 6\), under the long division bracket, and the divisor, \(4x + 2\), outside the bracket. Prepare to divide the first term of the dividend by the first term of the divisor.

Step 2: Divide the First Term

Divide the first term of the dividend, \(4x^2\), by the first term of the divisor, \(4x\). This gives you \(x\). Write \(x\) above the division bracket as part of the quotient.

Step 3: Multiply and Subtract

Multiply \(x\) by the divisor \(4x + 2\), giving \(4x^2 + 2x\). Subtract this result from the dividend: \((4x^2 - 10x + 6) - (4x^2 + 2x)\). This simplifies to \(-12x + 6\).

Step 4: Divide the Next Term

Divide the new first term of the resulting polynomial, \(-12x\), by the first term of the divisor, \(4x\). This gives \(-3\). Write \(-3\) as the next part of the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-3\) by the divisor \(4x + 2\), resulting in \(-12x - 6\). Subtract this from the current polynomial: \((-12x + 6) - (-12x - 6)\). This simplifies to \(12\), which is our remainder.

Key Concepts

Long Division

Long division is a systematic approach for dividing polynomials. This method, much like long division with numbers, helps in breaking down complex polynomials into simpler terms. To start, write the dividend, the polynomial being divided, inside the long division bracket, and the divisor, the polynomial you're dividing by, outside. The process involves repeatedly dividing and subtracting terms.

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result from the division step, write under the dividend, and subtract.
• The result becomes the new dividend to continue dividing.

It's all about tackling one term at a time, gradually simplifying the expression until division can't proceed without a remainder.

Quotient

In polynomial division, the quotient is the result of the division, equivalent to the answer you obtain in the top part of a division operation. You determine the quotient by dividing the leading term of your dividend by the leading term of the divisor. Each successful division provides a part of the quotient, accumulating until the division process is complete.

• Step-by-step, write each part of the quotient above the division bracket as you divide.
• The quotient provides the simplified expression of your original dividend after division.

Understanding the quotient is key to interpreting the division results, showcasing how much of the original polynomial can be expressed as multiples of the divisor.

Remainder

The remainder in polynomial division is similar to a remainder in arithmetic division. It is what is left when you've simplified as much as possible using whole numbers of the divisor. In the polynomial division, once you've performed the last division and subtraction, the remaining polynomial is your remainder.

• It represents the degree of the original dividend that isn't completely divisible by the divisor.
• A remainder can sometimes be zero, meaning the dividend is wholly divisible by the divisor.

In this exercise, the final remainder is 12, indicating that after extracting all possible multiples of the divisor from the dividend, 12 is what's left.

Algebraic Expressions

Algebraic expressions form the foundation of polynomial division. These are mathematical phrases consisting of variables, numbers, and operations (such as addition and multiplication). In polynomial division, you work with these expressions to break them down into parts—quotients and remainders.

• Variables, typically represented by letters like \(x\), are crucial to these expressions.
• Operations follow algebraic rules ensuring terms are simplified systematically.

Mastery of manipulating algebraic expressions is pivotal for understanding polynomial division, allowing you to see patterns and solve complex polynomial-related problems with ease.