Determining the Value of ( a ) for which ( x - 5 ) is a Factor of ( x^3 - 3x^2 ax - 10 )
To determine the value of ( a ) such that ( x - 5 ) is a factor of the polynomial ( x^3 - 3x^2 ax - 10 ), we can use the Factor Theorem. According to the Factor Theorem, if ( x - c ) is a factor of a polynomial ( P(x) ), then ( P(c) 0 ).
Application of the Factor Theorem
In this case, we set ( c 5 ) and evaluate the polynomial:
[P(x) x^3 - 3x^2 ax - 10]Now we substitute ( x 5 ) into ( P(x) ):
[P(5) 5^3 - 3(5^2) a(5) - 10]Calculating each term individually:
( 5^3 125 ) ( 3(5^2) 3 times 25 75 ) ( a(5) 5a )Substituting these values into ( P(5) ):
[P(5) 125 - 75 5a - 10]Simplifying this expression:
[P(5) 125 - 75 - 10 5a 40 5a]To ensure that ( x - 5 ) is a factor, we need ( P(5) 0 ):
[40 5a 0]Solving for ( a ):
[5a -40] [a -8]Therefore, the value of ( a ) for which ( x - 5 ) is a factor of ( x^3 - 3x^2 ax - 10 ) is ( boxed{-8} ).
Further Insights into the Factor Theorem
The Factor Theorem not only helps in finding the value of ( a ), but also tells us that ( x - 5 ) will be a factor of the polynomial if and only if the polynomial evaluated at ( x 5 ) equals zero. This is a powerful tool in algebraic problem-solving, as it provides a straightforward method for checking divisibility without performing polynomial division.
Examples and Practice
Understanding the Factor Theorem is crucial in various algebraic contexts. For instance, it can be applied to simplify polynomial expressions, solve polynomial equations, or factorize polynomials. It is particularly useful in scenarios where you need to determine the roots of a polynomial or to verify the existence of a particular factor.
Conclusion
The Factor Theorem is a fundamental concept in algebra that significantly simplifies the process of determining factors of a polynomial. By setting the polynomial equal to zero when the factor is substituted, we can easily find the unknown coefficients, making it a valuable tool for students and mathematicians alike. In this specific case, we determined that ( a -8 ) ensures that ( x - 5 ) is a factor of the given polynomial.
Further Reading
Factor Theorem and Polynomial Division: An in-depth exploration into the relationship between factor theorem and polynomial division, demonstrating the underlying principles and applications.
Algebraic Polynomials and Roots: A comprehensive guide to understanding the nature of polynomial roots and their significance in algebraic expressions.
The Role of the Factor Theorem in Solving Polynomial Equations: A detailed discussion on how the factor theorem can be used to solve complex polynomial equations efficiently.