Problem 51 - Algebra and Trigonometry

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

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

Step 1: Understanding the Factor Theorem

The Factor Theorem states that \(x-c\) is a factor of the polynomial \(P(x)\) if and only if \(P(c) = 0\). For this problem, we need to verify whether \(x-1\) is a factor of \(P(x)=x^3-3x^2+3x-1\) by checking if \(P(1)=0\).

Step 2: Substituting the value of c into P(x)

Substitute \(c=1\) into the polynomial and calculate \(P(1)\).\[P(1) = 1^3 - 3(1)^2 + 3(1) - 1\] Calculate step-by-step:- \(1^3 = 1\)- \(-3(1)^2 = -3\)- \(+ 3(1) = +3\)- \(- 1 = -1\)

Step 3: Calculating the result

Now simplify the expression:\[1 - 3 + 3 - 1 = 0\]Since \(P(1) = 0\), we conclude that \(x-1\) is a factor of the polynomial \(P(x)\).

Step 4: Conclusion

The calculation shows that \(P(1)=0\), confirming that \(x-1\) is indeed a factor of \(P(x)\). This is in line with the Factor Theorem which requires \(P(c) = 0\) for \(x-c\) to be a factor.

Key Concepts

Polynomial Division

Polynomial division is a process similar to dividing numbers, but in this case, we deal with polynomials instead of plain digits. Just like dividing numbers, the outcome is a quotient and sometimes a remainder. Let's break it down to make it simple:

• Consider a polynomial, which is an expression consisting of variables and coefficients.
• When dividing one polynomial by a simpler polynomial, like \(x-c\), we want to see if the expression divides evenly. If it does, the divisor is known as a factor.

We use a method similar to long division to carry this out, and the aim is to reduce the polynomial step by step. When dealing with the Factor Theorem, knowing that \(P(c)=0\) confirms that there is no remainder, making \(x-c\) a precise factor. This not only helps in simplifying polynomials but also in identifying roots more efficiently.

Roots of Polynomial

Roots of a polynomial are essentially the values of the variable that make the polynomial equal to zero. If \(x=r\) is a root of the polynomial \(P(x)\), this means substituting \(r\) into the polynomial makes it zero: \(P(r)=0\).

• Finding the roots helps in understanding the behavior of the polynomial graphically.
• Each root corresponds to a point where the graph intersects the x-axis.

The connection with the Factor Theorem comes into play because if \(P(c) = 0\), it means \(x-c\) is a factor of the polynomial, and thereby, \(c\) is a root.

Factorization of Polynomials

Factorization of polynomials involves breaking down a complex polynomial into simpler components, which are the factors. These factors, when multiplied together, give the original polynomial.

• Identifying a factor through the Factor Theorem can simplify this process significantly.
• Once a factor, such as \(x-c\), is known, we can perform polynomial division to further break down the polynomial.
• This splitting allows for easier solutions and analysis, particularly when solving polynomial equations.

In our example, by confirming that \(x-1\) is a factor, it suggests a start to factorizing \(P(x)\). Factorization also aids in sketching polynomial graphs and identifying all possible roots.