How do you use synthetic substitution to find p(-4) for #p(x)=3x^3-2x^2+6x-4#?

1 Answer
Aug 5, 2015

Answer:

#color(red)(p(-4) = -252)#

Explanation:

#p(x) = 3x^3-2x^2+6x-4#

The Remainder Theorem states that when we divide a polynomial #f(x)# by #x-c# the remainder #R# equals #f(c)#.

We use synthetic substitution to divide #f(x)# by #x-c#, where #c = -4#.

Step 1. Write only the coefficients of #x# in the dividend inside an upside-down division symbol.

#|3" "-2" " "6" " " "color(white)(1)-4#
#|color(white)(1)#
#stackrel("—————————————)#

Step 2. Put the divisor at the left.

#color(red)(-4)|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|" "#
#" "" "stackrel("—————————————)#

Step 3. Drop the first coefficient of the dividend below the division symbol.

#-4|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|color(white)(1)#
#" "" "stackrel("—————————————)#
#" "" "color(red)(3)#

Step 4. Multiply the drop-down by the divisor, and put the result in the next column.

#-4|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|" " "color(white)(1)color(red)(-12)#
#" "" "stackrel("—————————————)#
#" "" "3#

Step 5. Add down the column.

#-4|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|" " "-12#
#" "" "stackrel("—————————————)#
#" "" "3" "color(white)(1)color(red)(-14)#

Step 6. Repeat Steps 4 and 5 until you can go no farther.

#-4|3" "-2" "color(white)(1)6" " "color(white)(1)-4#
#" "color(white)(1)|" " " " "-12" " 56" "color(white)(1)-248#
#" "" "stackrel("—————————————)#
#" "" "3" "-14color(white)(1)62" "color(red)(-252)#

The remainder is #-252#, so #p(-4) = -252#.

Check:

#3x^3-2x^2+6x-4 = 3(-4)^3-2(-4)^2+6(-4)-4 = 3(-64)-2(16) -24-4= -192-32-28=-252#