Question

Perform the indicated division and write your answers in the form P(x)/D(x) = Q(x) + R(x)/D(x) as shown in the following example;​

Answer 1

`\textsf{1.} \quad 5x+3-(3)/(x-4)`

`\textsf{2.} \quad 2x^2-4x+3`

`\textsf{3.} \quad x^3+3x^2-2x+4+(25)/(2x-5)`

Explanation:

Long Division Method of dividing polynomials

• Dividend: The polynomial which has to be divided.
• Divisor: The expression by which the divisor is divided.
• Quotient: The result of the division.
• Remainder: The part left over.

Divide the first term of the dividend by the first term of the divisor and put that in the answer.

Multiply the divisor by that answer, put that below the dividend and subtract to create a new polynomial.

Repeat until no more division is possible.

Write the solution as the quotient plus the remainder divided by the divisor.

Question 1

`\large \begin{array}{r}5x+3\phantom{)}\\x-4{\overline{\smash{\big)}\,5x^2-17x-15\phantom{)}}}\\{-~\phantom{(}\underline{(5x^2-20x)\phantom{-b)..}}\\3x-15\phantom{)}\\-~\phantom{()}\underline{(3x-12)\phantom{}}\\-3\phantom{)}\\\end{array}`

Solution

`(5x^2-17x-15) / (x-4)=(5x^2-17x-15)/(x-4)=5x+3-(3)/(x-4)`

Question 2

`\large \begin{array}{r}2x^2-4x+3\phantom{)}\\3x-2{\overline{\smash{\big)}\,6x^3-16x^2+17x-6\phantom{)}}}\\{-~\phantom{(}\underline{(6x^3-4x^2)\phantom{-bbbbbbbb.)}}\\-12x^2+17x-6\phantom{)}\\-~\phantom{()}\underline{(-12x^2+8x)\phantom{))))).}}\\9x-6\phantom{)}\\-~\phantom{()}\underline{(9x-6)\phantom{}}\\0\phantom{)}\end{array}`

Solution

`(6x^3-16x^2+17x-6) / (3x-2)=(6x^3-16x^2+17x-6)/(3x-2)=2x^2-4x+3`

Question 3

`\large \begin{array}{r}x^3+3x^2-2x+4\phantom{)}\\2x-5{\overline{\smash{\big)}\,2x^4+x^3-19x^2+18x+5\phantom{)}}}\\{-~\phantom{(}\underline{(2x^4-5x^3)\phantom{-bbbbbbbbbbb.bb.)}}\\6x^3-19x^2+18x+5\phantom{)}\\-~\phantom{()}\underline{(6x^3-15x^2)\phantom{))))bbbb..).}}\\-4x^2+18x+5\phantom{)}\\-~\phantom{()}\underline{(-4x^2+10x)\phantom{)))..}}\\8x+5\phantom{)}\\-~\phantom{()}\underline{(8x-20)}\\25\phantom{)}\end{array}`

Solution

`\begin{aligned}(2x^4+x^3-19x^2+18x+5) / (2x-5) & =(2x^4+x^3-19x^2+18x+5)/(2x-5)\\ & =x^3+3x^2-2x+4+(25)/(2x-5)\end{aligned}`