Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\frac{$$2x^5$$ - $$8x^4$$ + $$2x^3$$ + $$x^2$$}{$$2x^3 + 1$$} 2x3+12x5−8x4+2x3+x2

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 16

4장, 문제 16

Divide using long division. State the quotient, and the remainder, r(x). $\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}$

검증된 단계별 안내

Write the division in long division format, placing the dividend $2x^{5} + 9x^{3} + 3x^{2}$ under the division bar and the divisor $3x^{2} + 1$ outside.

Divide the leading term of the dividend, \$2x^{5}$, by the leading term of the divisor, $3x^{2}$, to find the first term of the quotient: $\frac{2x^{5}}{3x^{2}} = \frac{2}{3}x^{3}$.

Multiply the entire divisor $3x^{2} + 1$ by the term $\frac{2}{3}x^{3}$ and subtract the result from the dividend to find the new remainder.

Repeat the process: divide the leading term of the new remainder by \$3x^{2}$ to find the next term of the quotient, multiply the divisor by this term, and subtract again.

Continue this process until the degree of the remainder is less than the degree of the divisor. The quotient is the sum of all terms found, and the remainder is the final expression left.

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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.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps determine when to stop the division process, as the remainder must have a degree less than that of the divisor.

Quotient and Remainder in Polynomial Division

When dividing polynomials, the result consists of a quotient and a remainder. The quotient is the polynomial obtained from the division steps, and the remainder is the leftover polynomial with a degree less than the divisor, often expressed as r(x).

Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\(\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}\) 2x3+12x5−8x4+2x3+x2