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avatar+110206 
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6 hours ago
 #1
avatar+25508 
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1.

\(f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5\)
 

Without using long division, find the remainder \((=r)\)  when \(f(x)\)  is divided by \(x^2-1\).

 

Let \(x^2-1 = (x-1)(x+1)\)

 

\(\begin{array}{|lrclrcl|} \hline & f(x) &=& q(x)(x-1)(x+1) + r \quad | \quad \mathbf{r=ax+b} \\ & \mathbf{ f(x) } &=& \mathbf{ q(x)(x-1)(x+1) + ax+b } \\ \hline x=1:& \mathbf{ f(1) } &=& \mathbf{ q(1)(1-1)(1+1) + a*1+b } \\ & f(1)&=& q(1)*0 + a+b \\ & f(1)&=& a+b \\ &&& \small {f(1) = 1*1^{10}+5*1^9-8*1^8+7*1^7-1^6-12*1^5+4*1^4-8*1^3+12*1^2-5*1-5 }\\ &&& f(1) = 1+5-8+7-1-12+4-8+12-5-5 \\ &&& f(1) = -8+7+4-8-5 \\ &&& f(1) = 11-21 \\ &&& f(1) = -10 \\ & -10 &=& a+b \\ & \mathbf{a+b} &=& \mathbf{-10} \qquad (1) \\ \hline x=-1:& \mathbf{ f(-1) } &=& \mathbf{ q(-1)(-1-1)(-1+1) + a*(-1)+b } \\ & f(-1)&=& q(-1)*0 - a+b \\ & f(-1)&=& - a+b \\ &&& \small{f(-1) = 1*(-1)^{10}+5*(-1)^9-8*(-1)^8+7*(-1)^7-(-1)^6-12*(-1)^5+4*(-1)^4-8*(-1)^3+12*(-1)^2-5*(-1)-5} \\ &&& f(-1) = 1-5-8-7-1+12+4+8+12+5-5 \\ &&& f(-1) =12+4 \\ &&& f(-1) = 16 \\ & 16 &=& - a+b \\ & \mathbf{-a+b} &=& \mathbf{16} \qquad (2)\\ \hline \end{array}\)

 

\(\begin{array}{|lrcll|} \hline & \mathbf{a+b} &=& \mathbf{-10} \qquad (1) \\ & \mathbf{-a+b} &=& \mathbf{16} \qquad (2)\\ \hline (1)+(2): & a+b-a+b&=&-10+16 \\ & 2b&=& 6 \\ & \mathbf{b} &=& \mathbf{3} \\ \hline (1)-(2): & a+b-(-a+b) &=& -10-16 \\ & 2a &=& -26 \\ & \mathbf{a} &=& \mathbf{-13} \\ \hline & r &=& ax+b \\ & \mathbf{ r } &=& \mathbf{-13x+3} \\ \hline \end{array}\)

 

The remainder when \(f(x)\) is divided by \(x^2-1\) is \(\mathbf{-13x+3}\)

 

laugh

7 hours ago
 #1
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Every musician in the band plays either the fife or the drum (but not both). The ratio of fifers to drummers is 7 : 1. If 2 drummers join the group and 6 fifers quit, the ratio of fifers to drummers would be 3 : 1. How many musicians were in the original ensemble?  

 

                                                                  F         7  

The ratio of fifers to drummers is 7 : 1.     ––   =   ––          Therefore, F  =  7D                                               (1)  

                                                                  D         1  

 

                                                          

If 2 drummers join the group               D + 2          1

                                                           –––––   =   ––           Therefore 3D + 6  =  F – 6   thus   F  =  3D + 12    (2)

and 6 fifers quit                                   F – 6          3

 

 

Substitute 3D + 12 for F in equation (1)         3D + 12  =  7D   

                                                                                12  =  4D

                                                                                  3  =  D                           3 Drummers   

 

Substitute 3 for D in equation (1)                               F  =  7 • 3

                                                                                   F  =  21                        21 Fifers  

.

10 hours ago
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Aug 12, 2020, 12:10 AM
Aug 11, 2020
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