Problem 10 - Intermediate Algebra

Question

Fill in the blanks. For \(P(x)=x^{3}-4 x^{2}+x+6,\) suppose we know that \(P(3)=0\) Then _____ is a factor of \(x^{3}-4 x^{2}+x+6\).

Step-by-Step Solution

Answer

\((x - 3)\) is a factor of the polynomial.

Step 1: Understanding the Given Information

We are given a polynomial \(P(x) = x^3 - 4x^2 + x + 6\), and the statement that \(P(3) = 0\). This implies that 3 is a root of the polynomial.

Step 2: Applying the Factor Theorem

According to the factor theorem, if \(a\) is a root of a polynomial \(P(x)\), then \((x - a)\) is a factor of the polynomial. Here, since \(3\) is a root, it follows that \((x - 3)\) is a factor of \(P(x)\).

Key Concepts

Polynomial

A polynomial is a mathematical expression that consists of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. It's a fundamental concept in algebra that provides a simple way to express equations that contain multiple terms. Polynomials are often used in different areas of mathematics and applied sciences.

Key characteristics of a polynomial include:

• Each term in the polynomial is composed of a coefficient (a constant number) and a variable raised to a power.
• The degree of a polynomial is the highest power of the variable present in the polynomial.
• A polynomial can have multiple terms, such as the expression: \[P(x) = x^3 - 4x^2 + x + 6\].

This polynomial is of degree 3 because the highest power of the variable \(x\) is 3. Understanding polynomials is crucial as they lay the groundwork for more complex algebraic structures and functions.

Root of a Polynomial

The root of a polynomial is a value of the variable that makes the polynomial equal to zero. Finding the roots of a polynomial means finding the values for which the polynomial holds true as an equation set to zero. In simple terms, these are the values that when substituted into the polynomial make the entire expression zero.

If we consider the polynomial \[P(x) = x^3 - 4x^2 + x + 6\],we see that one of its roots is 3, as given by the information that \[P(3) = 0\].

Roots are significant in algebra because:

• They help determine the factors of the polynomial.
• They are critical in graphing functions as they indicate where the function intersects the x-axis.
• They are used to solve polynomial equations, especially when the polynomial represents real-world phenomena.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form \[(x - a)\].It's a quick and efficient technique mainly used to test potential roots and simplify division problems.

To use synthetic division, follow these steps:

• Write down the coefficients of the polynomial.
• Use the root value that makes the polynomial zero, such as 3 in the given problem.
• Perform synthetic division by bringing down the leading coefficient and multiplying it by the root value, adding this to the next coefficient, and repeating the process for all coefficients.

This method saves time and reduces the complexity of polynomial division by making calculations more straightforward. It's especially useful when checking if a specific linear factor divides the polynomial without a remainder. In this context, synthetic division would confirm that \[(x - 3)\] is indeed a factor of \[P(x)\].
Synthetic division works best with polynomials that have a root easily found or given, such as the scenario where \[P(3) = 0\]. Knowing these principles helps in simplifying complex algebraic problems.