Problem 58 - A Graphical Approach to College Algebra

Question

RELATING CONCEPTS For individual or group investigation (Exercises \(55-60\) ) The close relationships among \(x\) -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now, be apparent to you. Consider the graph of the polynomial function \(P(x)=x^{3}-2 x^{2}-11 x+12,\). If \(P(x)\) is divided by \(x-2,\) what is the remainder? What is \(P(2) ?\)

Step-by-Step Solution

Answer

Remainder is -10; \(P(2) = -10\).

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that if a polynomial \(P(x)\) is divided by \(x-a\), then the remainder of this division is \(P(a)\). This means if we want to find the remainder when \(P(x)\) is divided by \(x-2\), we simply need to evaluate \(P(2)\).

Step 2: Evaluate P(2)

Given the polynomial \(P(x) = x^3 - 2x^2 - 11x + 12\), substitute \(x = 2\) into the equation: \[ P(2) = (2)^3 - 2(2)^2 - 11(2) + 12. \] Simplify this to find \(P(2) = 8 - 8 - 22 + 12 = -10\).

Step 3: Conclude with Remainder and P(2)

Thus the remainder of \(P(x)\) divided by \(x-2\) is \(-10\), and \(P(2) = -10\). This reinforces the direct application of the Remainder Theorem.

Key Concepts

Polynomial Functions

Polynomial functions form the backbone of algebra and calculus. These functions, such as \( P(x) = x^3 - 2x^2 - 11x + 12 \), are expressed as sums of terms with varying powers of \( x \). Each term consists of a coefficient multiplied by a power of \( x \). In general, a polynomial in \( x \) can be written as:

• Constant term: The term without any \( x \) (e.g., \(+12\)).
• Linear term: The term with \( x^1 \) (e.g., \(-11x\)).
• Quadratic term: The term with \( x^2 \) (e.g., \(-2x^2\)).
• Cubic term and higher: Terms with \( x^3 \) and above (e.g., \(x^3\)).

Polynomial functions like these can describe various real-world phenomena, such as physics equations or curves in geometry. Understanding them is crucial for solving problems involving changes in these relationships.

x-intercepts

The concept of \(x\)-intercepts is linked to where a graph crosses the x-axis. These intercepts are the points where the function value is zero, i.e., \( f(x) = 0 \). Finding \(x\)-intercepts involves solving for \(x\) when \( P(x) \) is set to zero.For the polynomial \( P(x) = x^3 - 2x^2 - 11x + 12 \), the \(x\)-intercepts can be found by solving \( x^3 - 2x^2 - 11x + 12 = 0 \). This often involves factoring the polynomial or using algorithms if factoring isn't straightforward. - Each intercept corresponds to a real zero of the polynomial, highlighting the close relationship between these two concepts. - Knowing \(x\)-intercepts helps understand how the graph behaves and interacts with the axes.

Real Zeros

Real zeros of a polynomial function are the points where the polynomial equals zero. These are the solutions to the equation obtained when the polynomial is set to zero, similar to finding \(x\)-intercepts. Real zeros are important because they reveal where the graph of the polynomial touches or crosses the x-axis.For \( P(x) = x^3 - 2x^2 - 11x + 12 \), real zeros are values of \(x\) where \( P(x) = 0 \). These zeros can be rational or irrational, depending on the polynomial.

• Finding these zeros involves methods such as factoring or using the quadratic formula if applicable.
• More complex polynomials may require numerical solutions or graphing techniques to approximate zeros.

Understanding real zeros allows for predictions about the polynomial's behavior, assisting in graphing and comprehension of its roots.

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials, especially useful for dividing by linear factors. Unlike long division, synthetic division is quicker and involves fewer calculations. It is particularly practical for polynomials expressed in the form \( P(x) = x^3 - 2x^2 - 11x + 12 \).To divide \( P(x) \) by \( x - a \), you:

• Use the coefficients of the polynomial.
• Perform operations using \(a\), the value at which \(x\) is set in the divisor.

Synthetic division not only expedites finding the quotient and remainder of the division but also plays a key role when utilizing the Remainder Theorem. It simplifies processes in algebra, making polynomial problem-solving more efficient and manageable.