Question

using long division in exercises 19-26, use the long division algorithm to perform the division. 19. \\(\frac{x^2 - 8x + 15}{x - 3}\\) 20. \\(\frac{t^2 - 18t + 72}{t - 6}\\) 21. divide \\(21 - 4x - x^2\\) by \\(3 - x\\). 22. divide \\(5 + 4x - x^2\\) by \\(1 + x\\). 23. \\((12 - 17t + 6t^2) \div (2t - 3)\\) 24. \\((16s + 10s^2 - 8) \div (4 + 2s)\\) 25. \\(\frac{9x^3 - 3x^2 - 3x + 4}{3x + 2}\\) 26. \\(\frac{4y^3 + 12y^2 + 7y - 3}{2y + 3}\\)

Explanation:

19. Step1: Divide leading terms

$\frac{x^2}{x}=x$

19. Step2: Multiply divisor by $x$

$x(x-3)=x^2-3x$

19. Step3: Subtract from dividend

$(x^2-8x+15)-(x^2-3x)=-5x+15$

19. Step4: Divide new leading terms

$\frac{-5x}{x}=-5$

19. Step5: Multiply divisor by $-5$

$-5(x-3)=-5x+15$

19. Step6: Subtract to find remainder

$(-5x+15)-(-5x+15)=0$

20. Step1: Divide leading terms

$\frac{t^2}{t}=t$

20. Step2: Multiply divisor by $t$

$t(t-6)=t^2-6t$

20. Step3: Subtract from dividend

$(t^2-18t+72)-(t^2-6t)=-12t+72$

20. Step4: Divide new leading terms

$\frac{-12t}{t}=-12$

20. Step5: Multiply divisor by $-12$

$-12(t-6)=-12t+72$

20. Step6: Subtract to find remainder

$(-12t+72)-(-12t+72)=0$

21. Step1: Rearrange dividend

$-x^2-4x+21$

21. Step2: Divide leading terms

$\frac{-x^2}{-x}=x$

21. Step3: Multiply divisor by $x$

$x(3-x)=3x-x^2$

21. Step4: Subtract from dividend

$(-x^2-4x+21)-(-x^2+3x)=-7x+21$

21. Step5: Divide new leading terms

$\frac{-7x}{-x}=7$

21. Step6: Multiply divisor by $7$

$7(3-x)=21-7x$

21. Step7: Subtract to find remainder

$(-7x+21)-(-7x+21)=0$

22. Step1: Rearrange dividend

$-x^2+4x+5$

22. Step2: Divide leading terms

$\frac{-x^2}{x}=-x$

22. Step3: Multiply divisor by $-x$

$-x(1+x)=-x-x^2$

22. Step4: Subtract from dividend

$(-x^2+4x+5)-(-x^2-x)=5x+5$

22. Step5: Divide new leading terms

$\frac{5x}{x}=5$

22. Step6: Multiply divisor by $5$

$5(1+x)=5+5x$

22. Step7: Subtract to find remainder

$(5x+5)-(5x+5)=0$

23. Step1: Rearrange dividend

$6t^2-17t+12$

23. Step2: Divide leading terms

$\frac{6t^2}{2t}=3t$

23. Step3: Multiply divisor by $3t$

$3t(2t-3)=6t^2-9t$

23. Step4: Subtract from dividend

$(6t^2-17t+12)-(6t^2-9t)=-8t+12$

23. Step5: Divide new leading terms

$\frac{-8t}{2t}=-4$

23. Step6: Multiply divisor by $-4$

$-4(2t-3)=-8t+12$

23. Step7: Subtract to find remainder

$(-8t+12)-(-8t+12)=0$

24. Step1: Rearrange dividend

$10s^2+16s-8$

24. Step2: Divide leading terms

$\frac{10s^2}{2s}=5s$

24. Step3: Multiply divisor by $5s$

$5s(2s+4)=10s^2+20s$

24. Step4: Subtract from dividend

$(10s^2+16s-8)-(10s^2+20s)=-4s-8$

24. Step5: Divide new leading terms

$\frac{-4s}{2s}=-2$

24. Step6: Multiply divisor by $-2$

$-2(2s+4)=-4s-8$

24. Step7: Subtract to find remainder

$(-4s-8)-(-4s-8)=0$

25. Step1: Divide leading terms

$\frac{9x^3}{3x}=3x^2$

25. Step2: Multiply divisor by $3x^2$

$3x^2(3x+2)=9x^3+6x^2$

25. Step3: Subtract from dividend

$(9x^3-3x^2-3x+4)-(9x^3+6x^2)=-9x^2-3x+4$

25. Step4: Divide new leading terms

$\frac{-9x^2}{3x}=-3x$

25. Step5: Multiply divisor by $-3x$

$-3x(3x+2)=-9x^2-6x$

25. Step6: Subtract from new dividend

$(-9x^2-3x+4)-(-9x^2-6x)=3x+4$

25. Step7: Divide new leading terms

$\frac{3x}{3x}=1$

25. Step8: Multiply divisor by $1$

$1(3x+2)=3x+2$

25. Step9: Subtract to find remainder

$(3x+4)-(3x+2)=2$

26. Step1: Divide leading terms

$\frac{4y^3}{2y}=2y^2$

26. Step2: Multiply divisor by $2y^2$

$2y^2(2y+3)=4y^3+6y^2$

26. Step3: Subtract from dividend

$(4y^3+12y^2+7y-3)-(4y^3+6y^2)=6y^2+7y-3$

26. Step4: Divide new leading terms

$\frac{6y^2}{2y}=3y$

26. Step5: Multiply divisor by $3y$

$3y(2y+3)=6y^2+9y$

26. Step6: Subtract from new dividend

$(6y^2+7y-3)-(6y^2+9y)=-2y-3$

26. Step7: Divide new leading terms

$\frac{-2y}{2y}=-1$

26. Step8: Multiply divisor by $-1$

$-1(2y+3)=-2y-3$

26. Step9: Subtract to find remainder

$(-2y-3)-(-2y-3)=0$

Answer:

1. $x-5$
2. $t-12$
3. $x+7$
4. $-x+5$
5. $3t-4$
6. $5s-2$
7. $3x^2-3x+1+\frac{2}{3x+2}$
8. $2y^2+3y-1$