In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (4x^4−4x^2+6x)/(x−4)

Verified step by step guidance

1. Step 1: Set up the long division by writing the dividend $4x^4-4x^2+6x$ under the division symbol and the divisor $x-4$ outside.
2. Step 2: Divide the first term of the dividend $4x^4$ by the first term of the divisor $x$ to get the first term of the quotient, which is $4x^3$.
3. Step 3: Multiply the entire divisor $x-4$ by the first term of the quotient $4x^3$ and subtract the result from the original dividend.
4. Step 4: Bring down the next term from the dividend, if necessary, and repeat the process: divide the new leading term by $x$, multiply the divisor by this new term, and subtract.
5. Step 5: Continue this process until all terms have been brought down and divided, resulting in a final quotient and a remainder, $r(x)$.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It involves a process similar to numerical long division, where the leading term of the dividend is divided by the leading term of the divisor, and the result is multiplied by the entire divisor. This product is then subtracted from the original polynomial, and the process is repeated with the resulting polynomial until the degree of the remainder is less than that of the divisor.

Quotient and Remainder

In polynomial division, the quotient is the result of the division, representing how many times the divisor fits into the dividend. The remainder is what is left over after the division process, which cannot be divided by the divisor without resulting in a fraction. The relationship between the dividend, divisor, quotient, and remainder can be expressed as: Dividend = Divisor × Quotient + Remainder.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It is a crucial concept in polynomial division, as it determines the order of the polynomial and influences the division process. Understanding the degree helps in identifying when to stop the division process, as the remainder must have a degree lower than that of the divisor for the division to be complete.