Problem 46 - Algebra 2

Question

ACT/SAT What is the remainder when \(x^{3}-7 x+5\) is divided by \(x+3 ?\) \(\mathbf{A}-11\) \(\mathbf{B}-1\) \(\mathbf{C} 1\) \(\mathbf{D} 11\)

Step-by-Step Solution

Answer

The remainder is -1. Option B is correct.

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that if a polynomial \(f(x)\) is divided by \(x - a\), the remainder is \(f(a)\). In our case, \(f(x) = x^3 - 7x + 5\) and we are dividing by \(x+3\), which can be written as \(x - (-3)\). Therefore, we need to find \(f(-3)\).

Step 2: Substitute into the Polynomial

Substitute \(x = -3\) into the polynomial \(f(x) = x^3 - 7x + 5\). This will give us the remainder when the polynomial is divided by \(x + 3\).

Step 3: Calculate the Expression

Calculate \((-3)^3 - 7(-3) + 5\). Begin by computing \((-3)^3 = -27\), then \(-7(-3) = 21\). This gives us \(-27 + 21 + 5\).

Step 4: Simplify the Result

Simplify the expression \(-27 + 21 + 5\). First, combine \(-27 + 21\) to get \(-6\). Then, add \(+5\) to get \(-1\).

Key Concepts

Polynomial Division

Polynomial division is a method similar to long division, but it's used specifically for dividing polynomials. When we perform polynomial division, we are essentially finding out how one polynomial can be expressed as a quotient (with regard to another polynomial) and a remainder.
To better understand this concept, consider dividing polynomials where the divisor is a simple linear term like \(x + a\). When we divide, we want to see how many times the divisor can "fit" into the polynomial, similar to how numeric division works.
However, instead of numbers, we're dealing with terms that potentially have variables raised to various powers. The goal is often to see how much of the original polynomial is left over after taking out as many complete fits of the divisor as possible. This leftover part is our remainder. This process lays the groundwork for understanding the Remainder Theorem and simplifying polynomial evaluation.

Remainder

In the context of polynomial division, the remainder is what you get after performing division and can no longer continue the process because the remaining polynomial has a lower degree than the divisor.
The Remainder Theorem gives us a quick way to find this remainder without fully executing the division. It tells us that if we have a polynomial \(f(x)\) and we divide it by an expression of the form \(x - a\), the remainder of this division is simply \(f(a)\). In other words, by evaluating the polynomial at \(a\), we directly get the value of the remainder.
For example, for our given polynomial \(x^3 - 7x + 5\), to find the remainder when divided by \(x + 3\), we calculate \(f(-3)\). This approach simplifies the process and avoids lengthy division calculations.

Substitution

Substitution is quite simply the act of plugging a specific value into an expression or equation. In our scenario, we are using substitution to apply the Remainder Theorem effectively.
Here, we substitute the value \(x = -3\) into the polynomial \(f(x) = x^3 - 7x + 5\) to find the remainder when the polynomial is divided by \(x + 3\).
By substituting \(x = -3\), we transform the polynomial from an expression containing \(x\) to a numeric computation. This involves replacing every instance of \(x\) with \(-3\). It simplifies the problem to simple arithmetic operations, allowing for an easier calculation of the important remainder value.

Polynomial Evaluation

Polynomial evaluation refers to the process of finding the value of a polynomial function at a certain point. This involves substituting a specific value for the variable in the polynomial expression and calculating the corresponding output.
For the polynomial \(f(x) = x^3 - 7x + 5\), evaluating \(f(-3)\) means substituting \(-3\) for every \(x\) in the polynomial and calculating the result.
Start by computing \((-3)^3\), which is \(-27\). Then calculate \(-7 \times (-3)\), resulting in \(21\). Finally, add all parts: \(-27 + 21 + 5\) to get \(-1\). The result of this evaluation gives us the remainder, thus efficiently applying the Remainder Theorem to solve our problem without traditional division effort.