Problem 53 - Precalculus Mathematics for Calculus

Question

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c\). $$P(x)=x^{3}-3 x^{2}+3 x-1, \quad c=1$$

Step-by-Step Solution

Answer

So, \(x-1\) is a factor of \(P(x)\) because \(P(1) = 0\).

Step 1: Understand the Factor Theorem

The Factor Theorem states that \(x-c\) is a factor of \(P(x)\) if and only if \(P(c) = 0\). In this exercise, we need to evaluate \(P(x) = x^3 - 3x^2 + 3x - 1\) at \(c = 1\) to determine if \(x-1\) is a factor.

Step 2: Substitute c into P(x)

Substitute \(c = 1\) into the polynomial \(P(x)\). This means computing \(P(1)\).\[P(1) = 1^3 - 3(1)^2 + 3(1) - 1\]

Step 3: Simplify the Expression

Compute each term in the expression \(P(1)\):- \(1^3 = 1\)- \(-3(1)^2 = -3\)- \(+3(1) = 3\)- \(-1 = -1\)Combine all the terms to simplify:\[P(1) = 1 - 3 + 3 - 1 = 0\]

Step 4: Analyze the Result

Since \(P(1) = 0\), according to the Factor Theorem, \(x-1\) is a factor of \(P(x) = x^3 - 3x^2 + 3x - 1\). Thus, substituting \(x = 1\) resulted in \(0\), confirming the theorem's conditions.

Key Concepts

Polynomial Division

When dealing with polynomials, one useful method for breaking them down is polynomial division. This technique allows us to divide one polynomial by another to obtain a quotient and possibly a remainder. It's similar to long division with numbers but applies to algebraic expressions.
One main goal of polynomial division is to simplify expressions or to factor polynomials, which are crucial tasks in algebra.

• First, set up the division by writing the dividend (the polynomial you are dividing) under the division bracket and the divisor (the polynomial you are dividing by) outside.
• Next, divide the leading term of the dividend by the leading term of the divisor. Write this result above the division bracket as part of the quotient.
• Then, multiply the entire divisor by this term and subtract the result from the original dividend.
• Bring down the next term of the dividend if necessary and repeat the steps until no terms are left to bring down.
• If the remainder is zero, then the divisor is a factor of the dividend.

This process is fundamental to understanding how polynomials can be expressed in simpler forms. It is especially useful when confirming factors identified by the Factor Theorem.

Roots of Polynomials

The roots, also known as zeros, of a polynomial are the values of the variable that make the polynomial equal to zero. Understanding the roots is essential as they provide insight into the behavior of polynomial functions. Here's how you can think about it:

• When a polynomial has a root at some value, say \(c\), it means that substituting \(c\) into the polynomial gives zero, i.e., \(P(c) = 0\).
• Each root corresponds to a factor of the polynomial. For example, if \(c\) is a root, then \(x-c\) is a factor of the polynomial.
• Finding all roots of a polynomial completely describes its solutions.
• Graphically, roots are the x-intercepts of the polynomial's graph, or points where the graph crosses the x-axis.

In our exercise, the root found using the Factor Theorem is \(c = 1\), showing that substituting \(x = 1\) yields zero. This confirms \(x - 1\) is a factor.

Factoring Polynomials

Factoring is the process of expressing a polynomial as the product of its simpler components or factors. When a polynomial is factored completely, it is expressed as a product of irreducible factors which cannot be further decomposed using integer coefficients.Here's how it connects to our exercise:

• The Factor Theorem helps identify factors quickly by checking roots. For example, if \(P(1) = 0\), then \(x-1\) is a factor.
• Once a factor is identified, polynomial division can be used to remove it and simplify the polynomial further.
• Other techniques such as grouping, special formulas, or synthetic division can also be employed to factor polynomials.
• The goal is to express polynomials in the simplest way to make solving equations easier or to analyze polynomial behavior.

In our case, verifying that \(x-1\) is a factor of \(P(x) = x^3 - 3x^2 + 3x - 1\) simplifies the factorization process, aiding in understanding the structure of the polynomial.