Problem 39 - College Algebra

Question

For the following exercises, factor the polynomials. $$ 125 a^{3}+343 $$

Step-by-Step Solution

Answer

The factored form is \((5a + 7)(25a^2 - 35a + 49)\).

Step 1: Recognize the Polynomial as a Sum of Cubes

The given polynomial expression is \( 125a^3 + 343 \). Notice that this is a sum of cubes, as \( 125a^3 \) is the cube of \( 5a \) and \( 343 \) is the cube of \( 7 \). Therefore, we are dealing with the expression \((5a)^3 + 7^3\).

Step 2: Apply the Sum of Cubes Formula

The sum of cubes formula is \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Here, \( a = 5a \) and \( b = 7 \). Substitute these into the formula.

Step 3: Substitute Values into the Formula

Substitute \( a = 5a \) and \( b = 7 \) into the formula: \[(5a + 7)((5a)^2 - 5a \cdot 7 + 7^2)\].

Step 4: Simplify the Expression Inside the Parentheses

Calculate \((5a)^2\), which is \(25a^2\), \(5a \cdot 7\), which is \(35a\), and \(7^2\), which is \(49\). Substitute these back into the expression to obtain:\[(5a + 7)(25a^2 - 35a + 49)\].

Step 5: Write the Final Factored Form

The polynomial \( 125a^3 + 343 \) is factored as \((5a + 7)(25a^2 - 35a + 49)\).

Key Concepts

Sum of Cubes

The sum of cubes is a special pattern in algebra that describes the addition of two cube numbers. When faced with a polynomial that resembles the form \( a^3 + b^3 \), it's important to recognize it as a sum of cubes quickly. In this specific exercise, we have the expression \( 125a^3 + 343 \). Decomposing it:

• \( 125a^3 \) is the cube of \( 5a \)
• \( 343 \) is the cube of \( 7 \)

We identify, therefore, that the polynomial is indeed a sum of cubes. When dealing with sums of cubes, you can employ the formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Understanding this pattern helps simplify and factor more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators. These expressions follow specific algebraic rules and can range from simple to complex. In the problem provided, we start with the polynomial expression \( 125a^3 + 343 \). Here:

• The variable is \( a \)
• Coefficients are numbers multiplying the variable or terms
• \( 125a^3 \) and \( 343 \) are terms of the expression

Working with algebraic expressions often involves identifying patterns or standard formulas, such as the sum of cubes. By breaking down and understanding each part, you can apply relevant formulas to simplify and factor the expression effectively. Always ensure you perform operations such as squaring or multiplying correctly to maintain the expression's integrity.

Polynomial Factorization

In mathematics, factorization involves breaking down a complex expression into simpler parts, often to solve equations or simplify problems. With polynomials, like our exercise's \( 125a^3 + 343 \), factorization can be crucial:

• Identify that you are working with a sum of cubes.
• Apply the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).

For the polynomial \( 125a^3 + 343 \), we substitute \( a = 5a \) and \( b = 7 \) into the formula, yielding:

• \((5a + 7)((5a)^2 - 5a \cdot 7 + 7^2)\)

After simplification, the terms inside the parentheses become \( 25a^2 - 35a + 49 \), resulting in the fully factored form \( (5a + 7)(25a^2 - 35a + 49) \). Mastering these steps enhances your ability to manipulate and solve higher degree polynomials efficiently. Understanding this process is a pivotal skill in algebra, as it simplifies functions and lays groundwork for more advanced math.