Problem 50 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine whether \(x-3\) is a factor of \(x^{4}+2 x^{3}-21 x^{2}+19 x-3\)

Step-by-Step Solution

Answer

Yes, x - 3 is a factor.

Step 1 - Apply the Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. Therefore, to determine whether x - 3 is a factor of the polynomial, we need to evaluate the polynomial at x = 3.

Step 2 - Substitute x = 3 into the Polynomial

Substitute x = 3 into the polynomial f(x) = x^4 + 2x^3 - 21x^2 + 19x - 3 and calculate the result: \[ f(3) = (3)^4 + 2(3)^3 - 21(3)^2 + 19(3) - 3 \] \[ f(3) = 81 + 54 - 189 + 57 - 3 \]

Step 3 - Simplify the Expression

Combine the terms: \[ 81 + 54 - 189 + 57 - 3 = 0 \]

Step 4 - Conclusion Based on the Factor Theorem

Since f(3) = 0, the Factor Theorem confirms that x - 3 is a factor of the polynomial x^4 + 2x^3 - 21x^2 + 19x - 3.

Key Concepts

Polynomial Factoring

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into a product of simpler polynomials, known as factors. Instead of dealing with a complicated polynomial, it's often easier to work with its factors.

For instance, consider the polynomial in the exercise: \(x^4 + 2x^3 - 21x^2 + 19x - 3\). To determine whether \(x-3\) is a factor, we use the Factor Theorem. Breaking down polynomials can help simplify finding solutions to equations and can reveal important properties of the polynomial.

• Factoring makes solving polynomial equations easier.
• It helps in identifying roots more efficiently.
• Applications include simplifying algebraic expressions and solving higher-degree equations.

Evaluation of Polynomials

Evaluating a polynomial at a given value involves substituting a specific number for the variable and calculating the result. This process is essential in verifying potential factors using the Factor Theorem.

To see if \(x-3\) is a factor of \(x^4 + 2x^3 - 21x^2 + 19x - 3\), we evaluate the polynomial at \(x=3\). Substitute \(3\) into the polynomial: \ f(3) = 3^4 + 2(3)^3 - 21(3)^2 + 19(3) - 3 \ f(3) = 81 + 54 - 189 + 57 - 3 = 0 \

• Substitute the variable with the given number.
• Calculate each term separately to avoid mistakes.
• Perform addition and subtraction to find the final value.

Since \(f(3)=0\), the polynomial evaluation confirms \(x-3\) is indeed a factor.

Roots of Polynomials

Roots of a polynomial are values of \(x\) that make the polynomial equal to zero. These are solutions to the polynomial equation.

If a polynomial \(f(x)\) can be factored as \((x-c)g(x)\), then \(x=c\) is a root of the polynomial. Applying the Factor Theorem, we found: \(f(3) = 0\). This means \(3\) is a root of the polynomial \(x^4 + 2x^3 - 21x^2 + 19x - 3\).

• Roots are critical for graphing polynomial functions.
• They help in understanding the behavior of polynomials.
• Finding the roots is the first step in factoring a polynomial completely.

Algebra II

Algebra II covers advanced topics beyond basic algebra, including polynomial functions, equations, and theorems like the Factor Theorem. These topics are essential for diving deeper into complex mathematical concepts.

In solving polynomial problems, students often use various theorems and methods to simplify and understand higher-degree polynomials. The Factor Theorem and polynomial factoring are just parts of this extensive field.

• Algebra II prepares students for more advanced math courses.
• It strengthens problem-solving and analytical skills.
• Topics include quadratic functions, polynomials, rational expressions, and more.
• Understanding these concepts is critical for success in calculus and other higher-level math classes.