Example 1:
Divide
\[ \dfrac{2x^3 + 3x^2 - x + 16}{x^2 + 2x - 3} \]
Solution:
The dividend \( 2x^3 + 3x^2 - x + 16 \) is a polynomial of degree 3. The divisor \( \; x^2 + 2x - 3 \; \) is a polynomial of degree \( 2 \). By laws of the exponents, we expect that the quotient is a polynomial of degree \( 1 \) or \( 0 \).
STEP 1: We first divide the term with highest power in the dividend \( \; 2x^3 \; \) by the term with the highest power in the divisor \( \; x^2 \; \)to obtain a quotient equal to \( \; 2x \; \) and organize all three terms as follows.
STEP 2: We next multiply the divisor \( \; x^2 + 2x - 3 \; \) by the quotient \( \; 2x \; \)and organize the result as follows
STEP 3: We next subtract the result of the multiplication from the dividend as follows
STEP 4: We now divide the term with the highest power in the subtraction result \( \; -x^2 \; \) by the term with the highest power in the divisor \( \; x^2 \;\) to obtain \( \; -1 \; \) and organize all terms as follows
STEP 5: We next multiply the divisor \( \; x^2 + 2x - 3 \; \) by \( \; -1 \; \) and organize all terms as follows
STEP 6: Subtract the result of the last multiplication from the term before it and organize the results as follows.
We now stop the process since the last term \( \; 7x + 13 \; \) has a degree smaller that that of the divisor \( \; x^2 + 2x - 3 \; \).
The result of the long division may be written as follows
\( \dfrac{2x^3 + 3x^2 - x + 16}{x^2 + 2x - 3} = 2x - 1 + \dfrac{7x + 13}{x^2 + 2x - 3} \)
or also as follows
\( 2x^3 + 3x^2 - x + 16 = (2x - 1)(x^2 + 2x - 3) + 7x + 13 \)
Vocabulary associated with the long division process
\( 2x^3 + 3x^2 - x + 16 \) is the dividend
\( x^2 + 2x - 3 \) is the divisor
\( 2x - 1 \) is the quotient
\( 7x + 13 \) is the remainder