[tex]4x^2\text{ + }\frac{19}{3}x\text{ + 11 + }\frac{26}{3x\text{ - 3}}[/tex]Explanation:[tex]\frac{12x^3+7x^2+14x\text{ }-7}{3x\text{ - 3}}[/tex]
To fill in the blank, we need to do the long division:
Since the division involves fraction, we will be dividing the numerator and denominator by 3 so it makes it easy to divide:
[tex]\begin{gathered} \frac{12x^3+7x^2+14x\text{ }-7}{3x\text{ - 3}}=\text{ }12x^3+7x^2+14x\text{ }-7\div3x\text{ - 3} \\ \frac{12x^3+7x^2+14x\text{ }-7}{3}\text{ }\div(\frac{3x\text{ - 3}}{3}) \\ \frac{12}{3}x^3+\frac{7}{3}x^2+\frac{14}{3}x\text{ }-\frac{7}{3}\div(\frac{3x}{3}\frac{-3}{3}) \\ =\text{ 4}x^3+\frac{7}{3}x^2+\frac{14}{3}x\text{ }-\frac{7}{3}\div(x-1) \\ =\text{ }\frac{\text{4}x^3+\frac{7}{3}x^2+\frac{14}{3}x\text{ }-\frac{7}{3}}{x\text{ - 1}} \end{gathered}[/tex][tex]\begin{gathered} \frac{12x^3+7x^2+14x\text{ }-7}{3x\text{ - 3}}=4x^2\text{ + }\frac{19}{3}x\text{ + 11 + }\frac{\frac{26}{3}}{x\text{ - 1}} \\ \frac{\frac{26}{3}}{x\text{ - 1}}\text{ = }\frac{26}{3}\div\text{ }(x-1)\text{ = }\frac{26}{3}\times\text{ }\frac{1}{x-1} \\ \frac{\frac{26}{3}}{x\text{ - 1}}\text{ =}\frac{26}{3(x\text{ - 1) }}\text{ = }\frac{26}{3x\text{ - 3}} \\ \\ T\text{he result:} \\ \frac{12x^3+7x^2+14x\text{ }-7}{3x\text{ - 3}}=4x^2\text{ + }\frac{19}{3}x\text{ + 11 + }\frac{26}{3x\text{ - 3}} \end{gathered}[/tex]
completing the statement:
[tex]\begin{gathered} \text{Begin the process by dividing }12x^3+7x^2+14x\text{ }-7\text{ by }3x\text{ - 3}, \\ which\text{ obtains }4x^2\text{ + }\frac{19}{3}x\text{ + 11 + }\frac{26}{3x\text{ - 3}}\text{. } \\ \text{Write this result above the quotient in the dividend} \\ \\ \text{obtains }4x^2\text{ + }\frac{19}{3}x\text{ + 11 as quotient and }26\text{ as remainder} \\ \end{gathered}[/tex]
