Problem 16 - Precalculus: Functions and Graphs

Question

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). $$f(x)=x^{4}-3 x^{3}-2 x^{2}+5 x+6 ; \quad c=2$$

Step-by-Step Solution

Answer

Yes, \(x-2\) is a factor of \(f(x)\) because \(f(2) = 0\).

Step 1: State the Factor Theorem

The Factor Theorem states that \(x-c\) is a factor of a polynomial \(f(x)\) if and only if \(f(c) = 0\). In this exercise, we need to find whether \(x-2\) is a factor of the given polynomial \(f(x)\) by evaluating \(f(2)\).

Step 2: Substitute and Evaluate f(c)

Substitute \(c = 2\) into the polynomial \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\). Calculate \(f(2)\):\[f(2) = (2)^4 - 3(2)^3 - 2(2)^2 + 5(2) + 6\]

Step 3: Calculate Each Term

Break down and calculate each term separately:- \((2)^4 = 16\)- \(-3(2)^3 = -24\)- \(-2(2)^2 = -8\)- \(5(2) = 10\)- \(+6 = 6\)

Step 4: Sum the Results

Add all the calculated terms together to find \(f(2)\):\[f(2) = 16 - 24 - 8 + 10 + 6\]

Step 5: Simplify the Expression

Simplify the expression by performing the arithmetic:\[f(2) = (16 - 24) + (10 - 8) + 6 = -8 + 2 + 6 = 0\]

Step 6: Conclusion

Since \(f(2) = 0\), according to the Factor Theorem, \(x-2\) is a factor of \(f(x)\).

Key Concepts

Polynomial

A polynomial is an algebraic expression made up of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They serve as fundamental building blocks in mathematics, representing a vast range of problems and scenarios.
Examples of polynomials include:

• Quadratic polynomials like \(ax^2 + bx + c\)
• Cubic polynomials such as \(ax^3 + bx^2 + cx + d\)
• Higher-degree polynomials like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)

A key feature of polynomials is their degree, which is the highest power of the variable. For instance, in the polynomial \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\), the degree is 4 because the highest power of \(x\) is 4. Polynomials are used to model a wide variety of scientific and engineering problems and serve as the foundation for calculus and algebra.

Polynomial Roots

The roots of a polynomial are the values of \(x\) which make the polynomial equal to zero. These are also known as "zeros" or "solutions." Finding the roots of a polynomial is essentially solving the equation \(f(x) = 0\).
Identifying the roots of a polynomial is important because:

• They indicate where the graph of the polynomial crosses the x-axis.
• Roots are essential in understanding the behavior of the polynomial function.
• They are critical in engineering and physics for solving real-world problems.

In the context of the Factor Theorem, if \(f(c) = 0\), then \(x - c\) is a factor of the polynomial. For the polynomial \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\), when we substituted \(c = 2\), we found \(f(2) = 0\), confirming that \(x=2\) is a root, and thus \(x-2\) is a factor.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form \(x-c\). It is faster and more efficient than traditional long division when working with polynomials and is often used when applying the Factor Theorem.
The process of synthetic division involves:

• Writing down the coefficients of the polynomial.
• Using the root \(c\) as a divisor.
• Performing a sequence of multiplications and additions to find the quotient and remainder.

For instance, to verify if \(x-2\) is a factor of \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\), you could use synthetic division. If it results in a remainder of 0, \(x-2\) is a factor. This method complements the Factor Theorem by providing a systematic approach to revealing polynomial factors efficiently.