How do you evaluate #f(x)=-x^5-4x^3+6x^2-x# at x=-2 using direct substitution and synthetic division?

Question

Christian Antunez · Accepted Answer

#f(-2)=90# for details please see below.

To evaluate #f(x)=-x^5-4x^3+6x^2-x# at x=-2 using direct substitution, we just put value of #x# in the polynomial. Hence, #f(-2)=-(-2)^5-4(-2)^3+6(-2)^2-(-2)# = #-(-32)-4(-8)+6xx4+2# = #32+32+24+2=90# Before we talk about synthetic division, one thing regarding remainder theorem which states that when a polynomial #f(x)# is divided by #x-a#, the remainder is #f(a)#. Hence, to find #f(-2)#, we should divide #f(x)# by #x-(-2)=x+2#, the remainder so obtained is #f(-2)#. Hence let us divide #f(x)=-x^5-4x^3+6x^2-x# by #x+2# using synthetic division method. One Write the coefficients of #x# in the dividend inside an upside-down division symbol. We are using #0# as coefficient of #x^4# and constant term. #color(white)(1)|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0# #color(white)(1)|" "color(white)(X)# #" "stackrel("——————————————————------)# Two As #x+2=0# gives #x=-2# put #-2# at the left. #-2|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0# #color(white)(xx)|" "color(white)(XX)# #" "stackrel("—————————————---------)# Three Drop the first coefficient of the dividend below the division symbol. #-2|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0# #color(white)(xx)|" "color(white)(X)# #" "stackrel("—————————————---------)# #color(white)(xx)|color(white)(xX)color(red)(-1)# Four Multiply the result by the constant, and put the product in the next column. #-2|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0# #color(white)(xx)|" "color(white)(xxXxx)2# #" "stackrel("————————————---------—)# #color(white)(xx)|color(white)(X)color(blue)-1# Five Add down the column. #-2|color(white)(X)1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0# #color(white)(xx)|" "color(white)(xXx)2# #" "stackrel("—————————————---------)# #color(white)(xx)|color(white)()color(blue)-1color(white)(X1)color(red)2# Six Repeat Steps Four and Five until you can go no farther. #-2|color(white)(X)1" "color(white)(X)0color(white)(X)-4" "" "6" "" "-1" "" "0# #color(white)(xx)|" "color(white)(Xxx)2color(white)(xx)-4color(white)(xxx)16color(white)(xxx)-44color(white)(xxx)90# #" "stackrel("———--------——————————)# #color(white)(xx)|color(white)()color(blue)-1color(white)(xxx)color(red)2color(white)(X)color(red)-8color(white)(XX)color(red)22color(white)(XX)color(red)-45color(white)(xxx)90# Hence, Quotient is #-x^4+2x^3-8x^2+22x-45# and remainder is #90# an hence #f(-2)=90#

James Allen · Answer

undefined To evaluate the function $ f(x) = -x^5 - 4x^3 + 6x^2 - x $ at $ x = -2 $ using direct substitution, follow these steps:

1. Substitute $ x = -2 $ into the function.
2. Evaluate the expression.

Direct Substitution:

$$ f(-2) = -(-2)^5 - 4(-2)^3 + 6(-2)^2 - (-2) $$
$$ f(-2) = -(-32) - 4(-8) + 6(4) + 2 $$
$$ f(-2) = 32 + 32 + 24 + 2 $$
$$ f(-2) = 90 $$

To evaluate the function using synthetic division, follow these steps:

1. Set up the coefficients of the polynomial in descending order of powers of $ x $.
2. Perform synthetic division with $ x = -2 $.
3. Evaluate the result.

Coefficients of $ f(x) = -x^5 - 4x^3 + 6x^2 - x $:
$$ -1, 0, -4, 6, 0, -1 $$

Synthetic Division:

$$ \begin{array}{c|rrrrr} -2 & -1 & 0 & -4 & 6 & 0 & -1 \ \hline & & 2 & -4 & 16 & -44 & 88 \ \end{array} $$

The result of the synthetic division is 88.

Therefore, $ f(-2) = 88 $.