# How do you use synthetic division to divide 5x^77 - 4x^60 + 2x^43 + 6x^30 - 4x^19 + 2 divided by x - 1?

Nov 15, 2015

That's going to take you forever.

#### Explanation:

Here's a link to a great video explaining Synthetic Division Synthetic Division-Khan Academy. Also, whoever assigned you that problem must wish you hell.

Nov 15, 2015

Also, use Wolfram Alpha to check your work ...

#### Explanation:

$5 {x}^{76} + 5 {x}^{75} + 5 {x}^{74} + 5 {x}^{73} + 5 {x}^{72} + 5 {x}^{71} + 5 {x}^{70} + 5 {x}^{69} + 5 {x}^{68} + 5 {x}^{67} + 5 {x}^{66} + 5 {x}^{65} + 5 {x}^{64} + 5 {x}^{63} + 5 {x}^{62} + 5 {x}^{61} + 5 {x}^{60} + {x}^{59} + {x}^{58} + {x}^{57} + {x}^{56} + {x}^{55} + {x}^{54} + {x}^{53} + {x}^{52} + {x}^{51} + {x}^{50} + {x}^{49} + {x}^{48} + {x}^{47} + {x}^{46} + {x}^{45} + {x}^{44} + {x}^{43} + 3 {x}^{42} + 3 {x}^{41} + 3 {x}^{40} + 3 {x}^{39} + 3 {x}^{38} + 3 {x}^{37} + 3 {x}^{36} + 3 {x}^{35} + 3 {x}^{34} + 3 {x}^{33} + 3 {x}^{32} + 3 {x}^{31} + 3 {x}^{30} + 9 {x}^{29} + 9 {x}^{28} + 9 {x}^{27} + 9 {x}^{26} + 9 {x}^{25} + 9 {x}^{24} + 9 {x}^{23} + 9 {x}^{22} + 9 {x}^{21} + 9 {x}^{20} + 9 {x}^{19} + 5 {x}^{18} + 5 {x}^{17} + 5 {x}^{16} + 5 {x}^{15} + 5 {x}^{14} + 5 {x}^{13} + 5 {x}^{12} + 5 {x}^{11} + 5 {x}^{10} + 5 {x}^{9} + 5 {x}^{8} + 5 {x}^{7} + 5 {x}^{6} + 5 {x}^{5} + 5 {x}^{4} + 5 {x}^{3} + 5 {x}^{2} + 5 x + \frac{7}{x - 1} + 5$

Source:
http://tinyurl.com/q96x3lo

Nov 15, 2015

You can use the remainder theorem to determine the remainder as:

$5 - 4 + 2 + 6 - 4 + 2 = 7$

As for the quotient, I would not use synthetic division...

#### Explanation:

Use:

${x}^{N} - 1 = \left(x - 1\right) {\sum}_{n = 0}^{N - 1} {x}^{n}$

Hence:

$5 {x}^{77} - 4 {x}^{60} + 2 {x}^{43} + 6 {x}^{30} - 4 {x}^{19} + 2$

$= 5 \left({x}^{77} - 1\right) - 4 \left({x}^{60} - 1\right) + 2 \left({x}^{43} - 1\right) + 6 \left({x}^{30} - 1\right) - 4 \left({x}^{19} - 1\right) + 7$

$= \left(x - 1\right) \left({\sum}_{n = 0}^{76} 5 {x}^{n} - {\sum}_{n = 0}^{59} 4 {x}^{n} + {\sum}_{n = 0}^{42} 2 {x}^{n} + {\sum}_{n = 0}^{29} 6 {x}^{n} - {\sum}_{n = 0}^{18} 4 {x}^{n}\right) + 7$

$= \left(x - 1\right) \left({\sum}_{n = 60}^{76} 5 {x}^{n} + {\sum}_{n = 43}^{59} {x}^{n} + {\sum}_{n = 30}^{42} 3 {x}^{n} + {\sum}_{n = 19}^{29} 9 {x}^{n} + {\sum}_{n = 0}^{18} 5 {x}^{n}\right) + 7$