Problem 34 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 8 a^{3}-4 a^{2}-12 a+16, \quad 4 $$

Step-by-Step Solution

Answer

Answer: The missing factor is \(2 a^{3} - a^{2} - 3 a + 4\).

Step 1: Identify the given product and factor

The given product is \(8 a^{3}-4 a^{2}-12 a+16\) and the given factor is \(4\).

Step 2: Divide the polynomial by the given factor

To find the missing factor, divide the polynomial by the given factor. In this case, divide \(8 a^{3}-4 a^{2}-12 a+16\) by \(4\): $$ \frac{8 a^{3}-4 a^{2}-12 a+16}{4} $$

Step 3: Perform the division

Now, divide each term in the polynomial by the given factor: $$ \frac{8 a^{3}}{4} - \frac{4 a^{2}}{4} - \frac{12 a}{4} + \frac{16}{4} $$

Step 4: Simplify each term to get the missing factor

Simplify each term: $$ 2 a^{3} - a^{2} - 3 a + 4 $$

Step 5: State the missing factor

The missing factor is \(2 a^{3} - a^{2} - 3 a + 4\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the building blocks of algebra, much like words in a sentence. They consist of variables, numbers, and operations (like addition or subtraction). An easy way to identify algebraic expressions is to look for terms that include letters (variables), often representing unknowns or changing quantities. These expressions can be as simple as a single number or variable, or as complex as the polynomial in the exercise above, \[8a^3 - 4a^2 - 12a + 16\]. Each term of an algebraic expression can include:

• A coefficient, which is a number multiplying the variable (e.g., in \(8a^3\), 8 is the coefficient).
• A variable, represented by letters like \(a, b, x\), which stand for unknown values.
• An exponent, which indicates how many times the variable is multiplied by itself, as seen in \(a^3\), where the exponent is 3.

Algebraic expressions are added, subtracted, or multiplied together to form more complex expressions. Understanding them is crucial because they are the foundation for more advanced mathematical topics like polynomials.

Grasping Polynomials

Polynomials are a special class of algebraic expressions that play a significant role in mathematics. A polynomial is composed of multiple terms that are added or subtracted, where each term includes a variable raised to a non-negative integer power. In the exercise, \[8a^3 - 4a^2 - 12a + 16\] is a polynomial of degree 3 because the highest power of the variable \(a\) is 3.Characteristics of polynomials include:

• The degree is determined by the highest exponent of the variable.
• They are closed under addition, subtraction, and multiplication (meaning performing these operations on polynomials results in another polynomial).
• Polynomials can be classified by degree, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.

Understanding polynomials is key for manipulating and simplifying expressions, modeling real-world problems, and solving algebraic equations. They provide a framework for analyzing complex relationships!

The Art of Factorization

Factorization is the process of breaking down an expression into a product of its factors. Just as a whole number can be factored into prime numbers, a polynomial can be decomposed into simpler polynomials or constants. In the given exercise, the task was to factor the polynomial \[8a^3 - 4a^2 -12a + 16\]by using one known factor, 4. This is done by polynomial division.Factorization can be achieved through these steps:

• Identifying common factors which all terms of the expression share, simplifying the expressions by dividing each term by this factor.
• For more complex polynomials, methods like grouping, using the quadratic formula, or seeking patterns can help.
• Recognizing special forms such as difference of squares, perfect squares, and cubes helps in quick decomposition.

By mastering factorization, you can simplify complex algebraic expressions, solve polynomial equations, and enhance your understanding of various mathematical patterns and properties.