Question 2.4.1: Long Division of Polynomials Divide x² + 10x + 21 by x + 3.
Long Division of Polynomials
Divide x² + 10x + 21 by x + 3.
The following process illustrates how polynomial division is very similar to division of whole numbers.
x+3)\overline{x^2+10x+21} Arrange the terms of the dividend (x² + 10x + 21) and the divisor (x + 3) in descending powers of x.
x+3)\overset{x}{\overline{x^2+10x+21}} Divide x² (the first term in the dividend) by x (the first term in the divisor): \frac{x^2}{x}=x. Align like terms.
Multiply each term in the divisor (x + 3) by x, aligning terms of the product under like terms in the dividend.
Subtract x² + 3x from x² + 10x by changing the sign of each term in the lower expression and adding.
Bring down 21 from the original dividend and add algebraically to form a new dividend.
Find the second term of the quotient. Divide the first term of 7x + 21 by x, the first term of the divisor: \frac{7 x}{x}=7.
Multiply the divisor (x + 3) by 7, aligning under like terms in the new dividend. Then subtract to obtain the remainder of 0.
The quotient is x + 7. Because the remainder is 0, we can conclude that x + 3 is a factor of x² + 10x + 21 and
\frac{x^2+10 x+21}{x+3}=x+7.