Problem 114 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value of \(k\) such that \(x-3\) is a factor of \(x^{3}-k x^{2}+2 k x-12\).

Step-by-Step Solution

Answer

The value of \(k\) is 5. Therefore, \(x - 3\) is a factor of the cubic polynomial \(x^{3}-5 x^{2}+10 x -12\).

Step 1: Apply the Factor Theorem

The factor theorem states that \(x-a\) is a factor of a polynomial \(f(x)\) if \(f(a) = 0\). So in order to find value of \(k\) for which \(x-3\) is a factor, one must plug \(x = 3\) into the polynomial equation \(x^{3}-k x^{2}+2 k x -12\) and set it equal to zero, giving \(3^{3}-k \cdot 3^{2}+2 k \cdot 3 -12 = 0\).

Step 2: Solve the equation

Solving the equation \(3^{3}-k \cdot 3^{2}+2 k \cdot 3 -12 = 0\) gives \(27 - 9k + 6k - 12 = 0\), which simplifies to \(-3k + 15 = 0\). Solving for \(k\) gives \(k = 15 / 3 = 5\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured in a specific format. The general form is a sum of terms, each term being a coefficient multiplied by a variable raised to a nonnegative integer power. For example, in the polynomial \(x^{3}-k x^{2}+2 k x-12\), the variable is \(x\), and the coefficients include \(-k\), \(2k\), and \(-12\).

• Each term in a polynomial is called a 'monomial', like \(kx^2\) or \(2kx\).
• The degree of a polynomial is the highest power of the variable, which in this case is 3.
• Polynomials can have various operations like addition, subtraction, multiplication, and division performed on them.

Understanding polynomials is crucial to diving deeper into algebra and calculus, as they form the building blocks for more complex calculus operations like differentiation and integration. In our exercise, we dealt with a cubic polynomial, which is when the highest exponent of the variable is 3.

Finding Roots

Finding roots of polynomials is an essential skill in algebra. The roots of a polynomial are the solutions of the equation when the polynomial is set equal to zero. For example, if we have \(p(x)=0\), then the solutions for \(x\) are called the roots.

• The Factor Theorem helps us find roots by stating that \(x-a\) is a root if \(f(a) = 0\).
• For our problem, we found that \(x=3\) must be a root.

Using this information, we substitute \(x = 3\) into our polynomial, which is \(x^{3}-k x^{2}+2 k x-12\), to set it to zero and solve for \(k\). By knowing how to find roots, you gain the ability to solve and factorize polynomial equations effectively.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. The polynomial \(x^{3}-k x^{2}+2 k x-12\) is one such algebraic expression.

• They allow us to generalize arithmetic processes, which can be solved using algebra.
• Variables, like \(x\) and \(k\), represent unknown values or changing quantities.
• Each component of the expression is linked through operator symbols.

Algebraic expressions, especially polynomials, are integral in representing real-world scenarios mathematically. Understanding how to interpret and manipulate these expressions is vital as it paves the way to solve equations and understand the behavior of different mathematical models.