Question

When the polynomial \(m x^{3}-3 x^{2}+n x+2\) is divided by \(x+3,\) the remainder is -1 When it is divided by \(x-2,\) the remainder is \(-4 .\) What are the values of \(m\) and \(n ?\)

Short answer

The values of \(m\) and \(n\) are -14 and 59, respectively.

Step-by-step solution

Understand the Problem

Given a polynomial of the form \(P(x) = mx^3 - 3x^2 + nx + 2\). When divided by \(x+3\), the remainder is -1. When divided by \(x-2\), the remainder is -4. Need to find values of \(m\) and \(n\).

Apply the Remainder Theorem for \(x+3\)

According to the Remainder Theorem, if \(P(x)\) is divided by \(x + 3\), then \(P(-3) = -1\). Substitute \(-3\) in the polynomial and set the result to \(-1\). \[P(-3) = m(-3)^3 - 3(-3)^2 + n(-3) + 2 = -1\] Simplify to get the first equation: \[-27m - 27 - 3n + 2 = -1\]

Simplify the First Equation

Combine like terms in the first equation: \[-27m - 3n - 25 = -1\] Add 25 to both sides: \[-27m - 3n = 24\]

Apply the Remainder Theorem for \(x-2\)

According to the Remainder Theorem, if \(P(x)\) is divided by \(x - 2\), then \(P(2) = -4\). Substitute \(2\) in the polynomial and set the result to \(-4\). \[P(2) = m(2)^3 - 3(2)^2 + n(2) + 2 = -4\] Simplify to get the second equation: \[8m - 12 + 2n + 2 = -4\]

Simplify the Second Equation

Combine like terms in the second equation: \[8m + 2n - 10 = -4\] Add 10 to both sides: \[8m + 2n = 6\]

Solve the System of Equations

Solve the system of linear equations: \[-27m - 3n = 24\] and \[8m + 2n = 6\]. Multiply the second equation by 3 to align the coefficients of \(n\): \[24m + 6n = 18\]. Add the two equations to eliminate \(n\): \[ -27m + 24m = 24 + 18\] Simplify to find \(m\): \[ -3m = 42 \] \[ m = -14\]

Find \(n\)

Substitute \(m = -14\) into one of the original equations: \[ 8(-14) + 2n = 6 \] \[-112 + 2n = 6 \]. Add 112 to both sides: \[2n = 118\] Solve for \(n\): \[n = 59\]

Key concepts

Remainder Theorem

The Remainder Theorem is a handy tool in polynomial algebra. It states that if a polynomial \(P(x)\) is divided by \(x-a\), the remainder of the division is \(P(a)\).
This means that by substituting \(x = a\) into the polynomial function, the result is the remainder of the division.
Understanding this theorem allows us to quickly determine the remainder without performing the full division.
In our problem, we applied the Remainder Theorem twice:

• When the polynomial is divided by \(x+3\), the remainder is -1, so \(P(-3) = -1\).
• When the polynomial is divided by \(x-2\), the remainder is -4, so \(P(2) = -4\).

This gave us two equations which we used to find the unknowns \(m\) and \(n\). The simplicity of this theorem can make polynomial division much more efficient!

System of Equations

A system of equations is a collection of two or more equations with a common set of unknowns.
To find the values of these unknowns, we can solve the system either algebraically or graphically.
In our polynomial problem, we created two equations from the given remainders:

• \(-27m - 3n = 24\)
• \(8m + 2n = 6\)

We used algebraic methods to solve this system.
First, we simplified and aligned the equations for easier computation.
After aligning the coefficients of \(n\) by multiplying the second equation by 3, we added the two equations to eliminate \(n\).
This allowed us to solve for \(m\). Substituting \(m = -14\) back into one of the original equations gave us the value of \(n = 59\).
Thus, solving systems of equations can unlock the variables and provide the precise answers we need.

Polynomial Functions

Polynomial functions are expressions involving terms powered by whole numbers.
They take the form \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \, \text{...} \, + a_1 x + a_0\) where \(a_n, a_{n-1}, ..., a_0\) are constants.
The highest power of the variable \(x\) determines the degree of the polynomial.
In our problem, the polynomial \(P(x) = mx^3 - 3x^2 + nx + 2\) is a cubic polynomial (degree 3).
Polynomial functions can be added, subtracted, multiplied, and divided, just like other algebraic expressions.
Understanding polynomial functions' behavior, such as how changing coefficients affects the graph and roots, is essential.
Key properties of polynomials include:

• Continuity and smooth curves
• Predictable end behavior based on the leading coefficient and degree
• The Fundamental Theorem of Algebra, which states that a polynomial of degree \(n\) has \(n\) roots in the complex number system

Speed and precision in dealing with polynomial problems come from mastering these core concepts!