Problem 22 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 21 x^{2}+7 x-14,7 $$

Step-by-Step Solution

Answer

Answer: The other factor is 3x^2 + x - 2.

Step 1: Set up the polynomial division

Set up the polynomial division with the given product and the given factor: $$ \frac{21x^2 + 7x - 14}{7} $$

Step 2: Perform the division

Now, divide the polynomial by the given factor. We will do this by dividing each term of the polynomial by the given factor: $$ \frac{21x^2}{7} + \frac{7x}{7} - \frac{14}{7} $$

Step 3: Simplify the division

Simplify the division results for each term: $$ 3x^2 + x -2 $$

Step 4: Write down the other factor

The simplified expression we obtained represents the other factor. Therefore, the other factor is: $$ 3x^2 + x -2 $$

Key Concepts

Understanding Polynomials

Polynomials are algebraic expressions that include variables, coefficients, and exponents. Simply put, they are sums of multiple terms, where each term follows a set pattern. In a term of a polynomial, you usually have variables raised to whole number powers, multiplied by coefficients. For instance, in the polynomial expression \(21x^2 + 7x - 14\), the terms are:

• \(21x^2\): Here, \(21\) is the coefficient, \(x\) is the variable, and \(2\) is the exponent.
• \(7x\): The coefficient is \(7\), the variable is \(x\), and the exponent is \(1\).
• \(-14\): This is a constant term, often considered as having a variable with an exponent of \(0\).

Understanding that polynomials can have one or two or many terms is essential. They can take the form of a simple number, like \(5\), or a quadratic expression like \(x^2 + 2x + 1\). Each polynomial behaves according to its own degree, which is the highest exponent on the variable.

The Art of Factorization

Factorization is the process of breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. In algebra, this is particularly useful for solving equations and simplifying expressions. To factorize a polynomial like \(21x^2 + 7x - 14\), look for common factors in all terms. In this case, we identified \(7\) as a factor. Once divided by \(7\), each term simplifies:

• \(\frac{21x^2}{7} = 3x^2\)
• \(\frac{7x}{7} = x\)
• \(\frac{-14}{7} = -2\)

Thus, the factorization gives us the simplified polynomial \(3x^2 + x - 2\). Recognizing such common factors in polynomials makes equations easier to handle and solve. Through factorization, you can often spot and solve roots of polynomials and simplify complex expressions into manageable chunks.

Exploring Algebraic Expressions

Algebraic expressions are collections of terms, which may include variables, numbers, and arithmetic operations. They're foundational to understanding algebra, since they encapsulate relationships and can convey complex equations in a concise format.Consider an expression like \(21x^2 + 7x - 14\). Within it, multiplying or dividing terms involves handling numerals and variables logically. Algebraic expressions are not restricted to integers or whole numbers; coefficients and constants can include negative and positive integers, fractions, or even decimals.When working with algebraic expressions, remember:

• Operations like addition, subtraction, multiplication, and division can change the form of the expression but should preserve equality.
• Variables can represent unknown values and help express mathematical relationships.
• Algebraic expressions can be simplified, evaluated, or manipulated, providing useful insights into complicated equations.

Mastering these fundamentals aids in solving equations, modeling real-world situations, and understanding higher-level math concepts.