Exercise 2.4.4
Recall how we divide a polynomial such as \(2x^3-3x^2-11x+6\) (called the dividend or the numerator) by a polynomial of same or lower degree such as \(x-4\) (called the divisor or the denominator).
- The divisor goes first, followed by a \(\big)\)
- After the \(\big)\) goes the dividend with a horizontal line drawn above it.
- Divide the leading term of the divisor into the leading term of the dividend and place the result above the line over the dividend. This is the first term of the quotient.
- The term just placed above the dividend is multiplied by every term of the divisor and placed under the term of the dividend of corresponding degree. A line is drawn under these terms and the terms are subtracted from the dividend and placed below the line.
- The results of the subtraction in the previous step is treated as a new dividend and the process of the last two steps is repeated until the subtraction step leaves a polynomial of lower degree than the degree of the divisor. That last term obtained is the remainder. The polynomial above the line over the dividend is the quotient.
| | | | \(2x^2\) | \(+5x\) | \(+9\) |
| \(x\) | \(-4\) | \(\big)2x^3\) | \(-3x^2\) | \(-11x\) | \(+6\) |
| | | \(2x^3\) | \(-8x^2\) | | |
| | | | \(5x^2\) | \(-11x\) | \(+6\) |
| | | \(5x^2\) | \(-20x\) | |
| | | | \(9x\) | \(+6\) |
| | | | | \(9x\) | \(-36\) |
| | | | | | \(42\) |
So the quotient is \(q(x)=2x^2+5x+9\) and the remainder is \(r(x)=42\).