#g(x) = 9x^4-8x^2+9x-5#
The Remainder Theorem states that when we divide a polynomial #g(x)# by #x-c# the remainder #R# equals #f(c)#.
So, we use synthetic substitution to divide #g(x)# by #x-c#, where #c = -5#.
Step 1. Write only the coefficients of #x# in the dividend inside an upside-down division symbol.
#color(white)(Xll)|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5#
#color(white)(XX)|#
#color(white)(XX)stackrel("———————————————————)#
Step 2. Put the divisor #(-5)# at the left.
#color(red)(-5)|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5#
#color(white)(XX)|#
#color(white)(XX)stackrel("———————————————————)#
Step 3. Drop the first coefficient of the dividend below the division symbol.
#-5|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5#
#color(white)(XX)|#
#color(white)(XX)stackrel("———————————————————)#
#color(white)(Xll)|color(red)(9)#
Step 4. Multiply the drop-down by the divisor, and put the result in the next column.
#-5|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5#
#color(white)(Xll)|color(white)(Xl)color(red)(-45)#
#color(white)(XX)stackrel("———————————————————)#
#color(white)(Xll)|9#
Step 5. Add down the column.
#-5|9color(white)(XXl)0color(white)(|l)-8color(white)(XXXll)9color(white)(Xll)-5#
#color(white)(Xll)|color(white)(ll)-45#
#color(white)(XX)stackrel("———————————————————)#
#color(white)(Xll)|9color(white)(ll)color(red)(-45)#
Step 6. Repeat Steps 4 and 5 until you can go no farther
#-5|9color(white)(XXl)0color(white)(Il)-8color(white)(XXXll)9color(white)(Xll)-5#
#color(white)(Xll)|color(white)(ll)-45" "225color(white)(l)-1085color(white)(Xl)5380#
#color(white)(XX)stackrel("———————————————————)#
#color(white)(Xll)|9color(white)(l)-45" "217color(white)(l)-1076" "color(red)(5375)#
∴ #g(-5) = 5375#
Check:
#g(x) = 9x^4-8x^2+9x-5#
#g(-5) = 9(-5)^4 -8(-5)^2+9(-5)-5 = 9(625)-8(25)-45-5= 5625-200-50 = 5375#
It works!
Now use the same method to show that
#g(4) = 2207#
