Question #ad6cf

2 Answers
Feb 24, 2018

#2x-1#

Explanation:

Let the third factor be #ax+b#, so that the polynomial is
# P(x) = (ax+b)(x+2)(x-4)#

#P(0) = 8 implies (a times 0+b)(0+2)(0-4) = 8 implies b=-1#

#P(1) = -9 implies (a times 1+b)(1+2)(1-4) = -9 implies a+b=1 implies a=2#

The third factor is #2x-1#

Feb 24, 2018

# "The answer is:" \qquad \qquad \qquad 2 x - 1. #

Explanation:

# "As" \ \ P(x) \ \ "has the two factors" \ x + 2 \quad "and" \quad x - 4, "we can write:" #

# \ P(x) \ = \ ( x + 2 ) ( x - 4) Q(x), \quad "for some polynomial" \ Q(x). \ (I)#

# "The polynomial," \ Q(x), "is the quantity requested in the" #
# "problem, and so is the solution to the problem." #

# "Now, taking the degree of both sides of this equation, we get:" #

# \qquad \qquad \qquad \qquad \qquad "deg" P(x) \ = \ "deg"( ( x + 2 ) ( x - 4) Q(x) ). #

# "Recalling that:" \quad "deg"( A(x) cdot B(x) ) \ = \ "deg"A(x) + "deg"B(x), #
# "we get:" #

# \qquad \qquad "deg" P(x) \ = \ "deg"( x + 2 ) + "deg" ( x - 4) + "deg" Q(x). #

# \qquad :. \qquad \qquad \qquad \quad "deg" P(x) \ = \ 1 + 1 + "deg" Q(x). #

# "And recalling, from the given, that:" \quad "deg" P(x) \ = \ 3, "we get:" #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad 3 \ = \ 1 + 1 + "deg" Q(x). #

# \qquad \qquad :. \qquad \qquad \qquad \qquad \qquad 3 \ = \ 2 + "deg" Q(x). #

# \qquad \qquad :. \qquad \qquad \qquad \qquad \qquad \quad "deg" Q(x) \ = \ 1. #

# \quad :. \quad \ Q(x) \ = \ a x + b; \quad \ "for some constants" \ a, b, \ a != 0. \quad \ \ (II) #

# "Substituting the previous into eqn. (I) above, we get:" #

# \qquad \qquad \qquad \qquad \qquad \quad P(x) \ = \ ( x + 2 ) ( x - 4) ( a x + b ). \qquad \qquad \qquad \qquad \quad \ (III) #

# "We are also given:" \ \ P(0) = 8. #

# "So substituting" \ \ x = 0 \ \ "into eqn. (III), we get:" #

# \qquad \qquad \qquad \qquad \qquad \quad \ P(0) \ = \ ( 0 + 2 ) ( 0 - 4 ) ( a cdot 0 + b ) #

# \qquad \qquad \qquad \qquad \qquad \qquad \quad\quad \ 8 \ = \ ( 2 ) ( - 4 ) ( b ) #

# \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ \ 8 \ = \ -8 b. #

# \qquad \qquad :. \qquad \qquad \qquad \quad \ \ b \ = \ -1. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ (IV) #

# "We are also given:" \ \ P(1) = -9. #

# "So substituting" \ \ x = 1 \ \ "into eqn. (III), we get:" #

# \qquad \qquad \qquad \qquad \qquad \quad \ P(1) \ = \ ( 1 + 2 ) ( 1 - 4 ) ( a cdot 1 + b ) #

# \qquad \qquad :. \qquad \qquad \qquad \quad \ -9 \ = \ ( 3 ) ( - 3 ) ( a + b ) #

# \qquad \qquad :. \qquad \qquad \qquad \quad \ -9 \ = \ -9 ( a + b ). #

# "Now, using the value we found for" \ \ b \ \ "in eqn. (IV), we get:"#

# \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ -9 \ = \ -9 ( a - 1 ) #

# \qquad \qquad :. \qquad \qquad \quad \ \ a - 1 \ = \ 1 #

# \qquad \qquad :. \qquad \qquad \qquad \qquad \quad \ a \ = \ 2. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ (V) #

# "Now, using the values we found for" \ \ a, b \ \ "in eqns. (IV) and (V)," #
# "and substituting them into eqn. (II) for" \ \ Q(x), "we get:"#

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ Q(x) \ = \ 2 x - 1. #

# "This is the answer to the question." #