Problem 28 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+3 \text { a factor of } x^{3}+x^{2}-14 x-24 ? $$

Step-by-Step Solution

Answer

Yes, \( x+3 \) is a factor of \( x^3+x^2-14x-24 \) because \( f(-3) = 0 \).

Step 1: Understanding the Factor Theorem

The Factor Theorem states that a polynomial \( f(x) \) has a factor \( x-a \) if and only if \( f(a) = 0 \). This means we need to check if plugging \( x = -3 \) into the polynomial \( f(x) = x^3 + x^2 - 14x - 24 \) results in zero.

Step 2: Substitute \( x = -3 \) into the Polynomial

Substitute \( x = -3 \) into the polynomial \( f(x) = x^3 + x^2 - 14x - 24 \). Calculate each term separately: - \((-3)^3 = -27\)- \((-3)^2 = 9\)- \(-14(-3) = 42\)- Constant term is \(-24\).

Step 3: Evaluate the Expression

Now, sum the calculated results from each term: \[-27 + 9 + 42 - 24\]Calculate the total:\[-27 + 9 = -18\]\[-18 + 42 = 24\]\[24 - 24 = 0\]

Step 4: Apply the Factor Theorem Conclusion

Since substituting \( x = -3 \) into the polynomial results in zero, by the Factor Theorem, \( x + 3 \) is indeed a factor of the polynomial \( x^3 + x^2 - 14x - 24 \).

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler polynomials that can be multiplied together to get the original polynomial. This is somewhat similar to factoring numbers in arithmetic. For example, just as you can factor the number 12 into 2, 2, and 3 since 12 = 2 × 2 × 3, you can factor polynomials in a similar way.

Factoring polynomials is a key skill in solving equations, simplifying expressions, and understanding the structure of algebraic formulas. It simplifies the process of finding polynomial roots and helps in various applications in algebra. The factor theorem is a fundamental tool in polynomial factorization because it sets the conditions under which a polynomial can be factored using linear monic polynomials like \( x - a \).

• Identify potential factors of the polynomial.
• Use strategies like the Factor Theorem to test potential factors.
• Continue to factor completely until the polynomial is expressed as a product of irreducible polynomials.

Understanding how to factorize polynomials leads not just to solving them but also provides insights into their behavior and characteristics.

Remainder Theorem

The Remainder Theorem is a powerful tool in algebra that provides a quick way to determine the remainder of a polynomial when it is divided by a linear divisor. It states that when a polynomial \( f(x) \) is divided by \( x - a \), the remainder of this division is \( f(a) \).

This theorem is especially useful when you want to check if a particular linear expression is a factor of a polynomial, as it directly links to the Factor Theorem. If \( f(a) = 0 \), then \( x - a \) is a factor, because there's no remainder left when dividing.

The process generally involves:

• Substituting \( x = a \) into the polynomial \( f(x) \).
• Calculating the result to check if the remainder is zero.
• Concluding whether the divisor is a factor based on the remainder.

This approach is not only efficient but also provides a clear method to handle polynomial division and factorization.

Polynomials

Polynomials are expressions consisting of variables and coefficients, connected using arithmetic operations like addition, subtraction, multiplication, and positive integer exponentiation. They form a central part of algebra and higher mathematics.

Understanding polynomials starts with recognizing their basic forms and structures, such as:

• Terms in a polynomial, which are parts of the expression separated by addition or subtraction signs.
• The degree of a polynomial, which is the highest power of the variable present in it.
• Coefficients, which are the numerical factors of each term.

These elements play essential roles in operations like polynomial addition, subtraction, multiplication, division, and especially factorization. Exploring how these parts fit together and work allows students to manipulate and solve polynomial equations effectively.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that form constructs capable of representing complex mathematical ideas. They are foundational in the study of algebra.

Recognizing different components of algebraic expressions, like terms, coefficients, and variables, is vital in simplifying and manipulating them:

• Terms – these are segments of an expression separated by addition or subtraction.
• Coefficients – numbers that multiply variables inside algebraic expressions.
• Variables – symbols that represent unknown or changeable values.

Using operations such as addition, subtraction, multiplication, and division, algebraic expressions can be simplified or transformed to solve equations or represent mathematical relationships. Advanced skills in manipulating algebraic expressions are crucial for dealing with real-world problems and more sophisticated fields of math.