Final answer:
The value of b in the polynomial x³ - 6x² - bx + 4, which leaves a remainder of -32 when divided by (X - 3), is found to be -3 by applying the Remainder Theorem.
Step-by-step explanation:
The student's question pertains to polynomial division and the remainder theorem. In order to find the value of b in the polynomial x³ − 6x² − bx + 4, given that it leaves a remainder of −32 when divided by (X − 3), we can apply the Remainder Theorem. This theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is f(a). Following this:
- Substitute x with 3 in the polynomial, as that is the value which makes (X − 3) equal to zero.
- The resulting equation is 3³ − 6⋅(3)² − b⋅(3) + 4
- Simplify the equation and solve for the remainder, which is set as −32.
- So, 27 − 54 − 3b + 4 = −32, which simplifies to − 3b − 23 = −32.
- Solve for b to get b = -3.
Thus, the value of b in the polynomial is − 3.