Problem 52 - 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 \(c .\) \(P(x)=x^{3}+2 x^{2}-3 x-10, \quad c=2\)

Step-by-Step Solution

Answer

\( x-2 \) is a factor of \( P(x) \) because \( P(2) = 0 \).

Step 1: Understand the Factor Theorem

The Factor Theorem states that \( x-c \) is a factor of a polynomial \( P(x) \) if and only if \( P(c) = 0 \). Here, \( P(x) = x^3 + 2x^2 - 3x - 10 \) and we need to check if \( x-2 \) is a factor. Thus, substitute \( x = 2 \) into \( P(x) \) and see if it equals zero.

Step 2: Substitute \( c = 2 \) into \( P(x) \)

Calculate \( P(2) \) by substituting \( x = 2 \) in the polynomial: \[P(2) = (2)^3 + 2(2)^2 - 3(2) - 10\]This simplifies to:\[8 + 8 - 6 - 10\]

Step 3: Simplify the Expression

Now, simplify the expression from Step 2:\[8 + 8 = 16 \16 - 6 = 10 \10 - 10 = 0\]

Step 4: Analyze the Result

Since \( P(2) = 0 \), according to the Factor Theorem, this means that \( x-2 \) is a factor of \( P(x) = x^3 + 2x^2 - 3x - 10 \).

Key Concepts

Polynomial Division

Polynomial division is similar to long division, but it is used to divide polynomials instead of numbers. This method helps in breaking down a polynomial into simpler components, especially when you need to verify whether a polynomial can be divided by another polynomial evenly. In our original exercise, if you want to divide \( P(x) = x^3 + 2x^2 - 3x - 10 \) by \( x - 2 \), polynomial division would be the method to use. You'll align the terms of the polynomials in descending powers and proceed similarly as with long division, performing term-by-term subtraction until you reach the end. When the remainder is zero, it confirms that the divisor is a factor, aligning with our original solution where \( x - 2 \) divided perfectly into \( P(x) \) with no remainder.

Roots of Polynomials

The roots of a polynomial are the solutions to the equation \( P(x) = 0 \). In simpler terms, these are the values of \( x \) for which the polynomial equals zero. Finding roots is crucial because each root corresponds to a factor of the polynomial. According to the Factor Theorem, if substituting \( c \) into the polynomial \( P(x) \) results in zero, then \( x-c \) is a factor, and \( c \) is a root. In the given problem, substituting \( x = 2 \) into \( P(x) = x^3 + 2x^2 - 3x - 10 \) yields zero, indicating \( x = 2 \) is indeed a root. Identifying roots aids in further factorizing and solving polynomials effectively.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This process is vital for simplifying polynomial expressions and solving polynomial equations. Once you identify a root using the Factor Theorem, as we did with \( c = 2 \) for \( P(x) \), you have effectively begun the factorization process. The polynomial can now be expressed as \( (x - 2)(Q(x)) \), where \( Q(x) \) is the quotient found through polynomial division. Continued factorization might lead to even simpler expressions, making it easier to find additional roots or simplify algebraic expressions. This technique is essential in algebra for breaking complex problems into manageable parts.