Understanding the Factorization of (x^3 - bx^2 - cx - 6) Given (x - 3) as a Factor

The problem at hand involves a cubic polynomial f(x) x^3 - bx^2 - cx - 6, with a given factor x - 3. The Factor Theorem states that if a polynomial f(x) has a factor of the form x - a, then f(a) 0. In this case, we substitute x 3 into the polynomial and set the result equal to zero.

Application of the Factor Theorem

To use the Factor Theorem, we substitute (x 3) into the polynomial f(x) x^3 - bx^2 - cx - 6).

First, we write down the equation:

[f(3) 3^3 - b(3^2) - 3c - 6 0]

This simplifies to:

[27 - 9b - 3c - 6 0]

Clean up the terms:

[21 - 9b - 3c 0]

Further rearrangement gives:

[21 9b 3c]

[7 3b c]

From this, we can deduce the relationship between (b) and (c):

[c 7 - 3b]

Analysis and Conclusion

Without additional information about the values of (b) and (c), this is the most specific relationship we can establish. Given that:

[3b c 7]

Any pair of values for (b) and (c) that satisfy this equation will yield a polynomial with (x - 3) as a factor. For example, if (b 2), then:

[c 7 - 3(2) 1]

Alternatively, if (b 1), then:

[c 7 - 3(1) 4]

Therefore, the relationship (c 7 - 3b) is the solution to the problem.

Conclusion

To summarize, if (x - 3) is a factor of the polynomial (f(x) x^3 - bx^2 - cx - 6), then we can conclude that (c 7 - 3b). This relationship holds true for any valid values of (b) and (c).

Key Points to Remember:

The Factor Theorem is a powerful tool for determining factors of polynomials. The relationship between the coefficients in a polynomial with a known factor can be determined algebraically. Without further constraints, multiple pairs of (b) and (c) values can satisfy the given condition.

Understanding these concepts is crucial for solving polynomial equations and simplifying functions in algebra.

References and Further Reading

Math is Fun - Polynomials Khan Academy - Polynomial Factor Theorem Polynomial Factorisation - SlideShare

By following these steps and utilizing these resources, you can deepen your understanding of polynomial factorization and improve your skills in solving complex algebraic problems.