In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (12x^2+x−4)÷(3x−2)

Verified step by step guidance

1. Set up the long division by writing \(12x^2 + x - 4\) under the division symbol and \(3x - 2\) outside.
2. Divide the first term of the dividend \(12x^2\) by the first term of the divisor \(3x\) to get the first term of the quotient, which is \(4x\).
3. Multiply \(4x\) by the entire divisor \(3x - 2\) and subtract the result from the original dividend \(12x^2 + x - 4\).
4. Bring down the next term from the dividend to form a new polynomial and repeat the division process with the new polynomial.
5. Continue this process until the degree of the remainder is less than the degree of the divisor, and then state the quotient and the remainder.

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 divisor is used to divide the leading term of the dividend, and the result is multiplied back and subtracted from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.

Quotient and Remainder

In polynomial division, the quotient is the result of the division, representing how many times the divisor can fit into the dividend. The remainder is what is left over after the division process, which cannot be divided by the divisor anymore. According to the polynomial division algorithm, any polynomial can be expressed as the product of the divisor and the quotient, plus the 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 helps in identifying when to stop the division process. Understanding the degree allows for proper alignment of terms during division and ensures that the remainder is of a lower degree than the divisor.