Problem 29 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-3 \text { a factor of } 3 x^{3}-5 x^{2}-17 x+17 ? $$

Step-by-Step Solution

Answer

No, \(x - 3\) is not a factor of the polynomial.

Step 1: Recall the Factor Theorem

The Factor Theorem states that \(x - c\) is a factor of a polynomial \(P(x)\) if and only if \(P(c) = 0\). In this problem, we need to check if \(x - 3\) is a factor of the polynomial \(3x^3 - 5x^2 - 17x + 17\).

Step 2: Substitute \(x = 3\) into \(P(x)\)

To apply the Factor Theorem, substitute \(x = 3\) into the polynomial. Calculate \(P(3)\), where \(P(x) = 3x^3 - 5x^2 - 17x + 17\).

Step 3: Perform the Substitution Calculation

Compute \(P(3)\):\[P(3) = 3(3)^3 - 5(3)^2 - 17(3) + 17\]Simplify:\[= 3(27) - 5(9) - 51 + 17\]\[= 81 - 45 - 51 + 17 \]\[= 2\]

Step 4: Interpret the Result

Since \(P(3) = 2\) and not 0, \(x - 3\) is not a factor of the polynomial \(3x^3 - 5x^2 - 17x + 17\) according to the Factor Theorem.

Key Concepts

Understanding Polynomial Factors

Polynomial factors are expressions that can be multiplied together to obtain a polynomial. If you have a polynomial, say \(P(x)\), and you suspect that \(x-c\) might be a factor, the Factor Theorem is a quick way to test this. The theorem states that \(x-c\) is a factor if substituting \(c\) into the polynomial yields 0, meaning \(P(c) = 0\).

This concept is essential when you're trying to break down polynomials into simpler, more manageable pieces. Finding factors is a bit like looking for the pieces that fit together to form the puzzle that is the original polynomial. If a particular expression is a factor, it means that polynomial can be divided without a remainder by that expression.

When working with higher-degree polynomials, such as cubic ones, understanding factors is critical for solving equations and finding roots efficiently.

Exploring the Substitution Method

The substitution method is a straightforward technique often used with the Factor Theorem. It involves replacing the variable in a polynomial with a specific number to check if the result is zero. In our problem, we are substituting \(x = 3\) into the polynomial \(3x^3 - 5x^2 - 17x + 17\) to see if it vanishes.

Here's how it works:

• Take a suspected root like \(x = 3\).
• Replace every instance of \(x\) in the polynomial.
• Simplify the expression by performing arithmetic operations.

Since our calculation showed \(P(3) = 2\), not zero, it confirmed that \(x-3\) is not a factor of the given polynomial.

By executing this method step-by-step, we're essentially checking if the polynomial equals zero, which is a direct application of the Factor Theorem.

Digging Into Cubic Polynomials

Cubic polynomials are algebraic expressions of the form \(ax^3 + bx^2 + cx + d\), where \(a eq 0\). Our example \(3x^3 - 5x^2 - 17x + 17\) is a typical cubic polynomial.

Understanding cubic polynomials is crucial because they represent three-dimensional relationships. They are common in various fields such as physics, engineering, and computer graphics. These polynomials can have up to three solutions, or \