In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (3x²−2x+5)/(x−3)

검증된 단계별 안내

Identify the dividend and divisor. Here, the dividend is $3x^{2} - 2x + 5$ and the divisor is $x - 3$.

Set up the long division by writing $3x^{2} - 2x + 5$ under the division bar and $x - 3$ outside the division bar.

Divide the leading term of the dividend, \$3x^{2}$, by the leading term of the divisor, $x$, to get the first term of the quotient: $\frac{3x^{2}}{x} = 3x$.

Multiply the entire divisor $x - 3$ by \$3x$ and subtract the result from the dividend. This will give you a new polynomial.

Repeat the process: divide the leading term of the new polynomial by $x$, multiply the divisor by this term, subtract again, and continue until the degree of the remainder is less than the degree of the divisor.

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주요 개념

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Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting, and repeating until the degree of the remainder is less than the divisor.

Quotient and Remainder in Polynomial Division

When dividing polynomials, the quotient is the result of the division, and the remainder is what is left over when the division cannot continue. The remainder must have a degree less than the divisor. The division 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 expression. Understanding the degree is essential in polynomial division because the process continues until the remainder's degree is less than the divisor's degree, indicating the division is complete.

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (3x2−2x+5)/(x−3)