Problem 48 - Algebra for College Students

Question

For Problems \(45-48\), find the value(s) of \(k\) that makes the second polynomial a factor of the first. $$ x^{3}+4 x^{2}-11 x+k ; x+2 $$

Step-by-Step Solution

Answer

The value of \(k\) is \(-30\).

Step 1: Understanding the Problem

We need to find the value of \(k\) such that \(x+2\) is a factor of the polynomial \(x^{3} + 4x^{2} - 11x + k\). This means that when we substitute \(-2\) into the polynomial, the result should be zero.

Step 2: Apply the Factor Theorem

According to the factor theorem, if \(x+2\) is a factor of the polynomial, then \(f(-2) = 0\), where \(f(x) = x^{3} + 4x^{2} - 11x + k\). So we substitute \(-2\) into the polynomial and set it equal to zero.

Step 3: Substitute and Solve for k

Substitute \(-2\) into the polynomial: \(f(-2) = (-2)^{3} + 4(-2)^{2} - 11(-2) + k = 0\). Simplifying, we have:\[-8 + 16 + 22 + k = 0\]Combine the constants:\[30 + k = 0\]Solve for \(k\):\[k = -30\].

Key Concepts

Polynomial Division

Polynomial division is a method to divide one polynomial by another, much like dividing numbers in arithmetic. It's particularly useful when you're trying to factor polynomials or simplify complex algebraic expressions. Let's breakdown the division process:

• Start by organizing the polynomials in descending order of their degrees.
• Divide the first term of the dividend by the first term of the divisor, write the result above the division bar.
• Multiply the entire divisor by the term you found, and subtract it from the dividend.
• Bring down the next term, and repeat the process until you can't divide any further.

This method is especially useful for checking if a polynomial like \(x+2\) is a factor of another polynomial, as in our given exercise. If there is no remainder, it confirms that the divisor is a factor.

Roots of Polynomials

The roots of a polynomial are the values of \(x\) that make the polynomial equal zero. They are the solutions to the polynomial equation. Identifying these roots is crucial since they represent the points where a polynomial graph intersects the x-axis.To find the roots:

• Set the polynomial equal to zero.
• Solve the equation using factoring, the quadratic formula, or by graphing.
• Use the Factor Theorem, which tells us that \(x-a\) is a factor of a polynomial if substituting \(a\) into the polynomial yields zero.

In this exercise, by using the factor theorem, we determined that \(-2\) is a root when \(k=-30\). This means that for \(x+2\) to be a factor, the polynomial must equal zero when \(x=-2\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the backbone of algebra, allowing us to formulate and solve equations.When dealing with algebraic expressions:

• You might need to simplify them by combining like terms.
• Substitute values into variables to evaluate the expression's result.
• Manipulate them to align with conditions such as when solving for a variable.

In our exercise, the polynomial \(x^3 + 4x^2 - 11x + k\) is an algebraic expression. We sought to find the value of \(k\) that makes the linear factor \(x+2\) divide it completely, such that the expression simplifies exactly to zero at the particular value of \(x=-2\). This condition uniquely determined \(k=-30\).