Certainly! To solve for the value of [tex]\( a \)[/tex] in the polynomial [tex]\( q(z) = z^3 - 4z + a \)[/tex] given that it leaves a remainder of 5 when divided by [tex]\( z - 3 \)[/tex], we can use the Remainder Theorem.
The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( q(z) \)[/tex] by [tex]\( z - c \)[/tex] is [tex]\( q(c) \)[/tex].
Given that [tex]\( q(z) \)[/tex] leaves a remainder of 5 when divided by [tex]\( z - 3 \)[/tex], this implies:
[tex]\[ q(3) = 5 \][/tex]
Now we need to substitute [tex]\( z = 3 \)[/tex] into the polynomial [tex]\( q(z) \)[/tex] and set it equal to 5:
[tex]\[ q(3) = 3^3 - 4(3) + a \][/tex]
Calculate [tex]\( q(3) \)[/tex] step by step:
[tex]\[ q(3) = 3^3 - 4(3) + a \][/tex]
[tex]\[ q(3) = 27 - 12 + a \][/tex]
[tex]\[ q(3) = 15 + a \][/tex]
According to the Remainder Theorem:
[tex]\[ 15 + a = 5 \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ a = 5 - 15 \][/tex]
[tex]\[ a = -10 \][/tex]
Thus, the value of [tex]\( a \)[/tex] is [tex]\( -10 \)[/tex].
