Problem 42 - College Algebra

Question

For the following exercises, factor the polynomials. $$ 729 q^{3}+1331 $$

Step-by-Step Solution

Answer

The factorization is \((9q + 11)(81q² - 99q + 121)\).

Step 1: Recognize as Sum of Cubes

The polynomial \( 729 q^3 + 1331 \) is a sum of two cubes since \( 729 = 9^3 \) and \( 1331 = 11^3 \). This means the expression can be written as the sum of cubes: \( (9q)^3 + 11^3 \).

Step 2: Use Sum of Cubes Formula

Recall that the sum of cubes formula is given by: \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \). Here, identify \( a = 9q \) and \( b = 11 \).

Step 3: Substitute into the Formula

Substitute \( a = 9q \) and \( b = 11 \) into the sum of cubes formula:\((9q + 11)((9q)^2 - (9q)(11) + 11^2)\).

Step 4: Simplify the Factored Form

Simplify the expressions:- \((9q)^2 = 81q^2\)- \((9q)(11) = 99q\)- \(11^2 = 121\)Thus, the expression is: \((9q + 11)(81q^2 - 99q + 121)\).

Key Concepts

Sum of Cubes

In algebra, recognizing patterns can simplify complex problems. One such pattern is the **sum of cubes**. It refers to expressing a polynomial as the sum of two cubes. Knowing this allows us to use a specific formula to factor it. Let's take the polynomial from our example:

• The expression is 729\(q^3\) which equals \((9q)^3\), meaning 9 is cubed.
• The number 1331 is \(11^3\), indicating 11 is cubed.

In this case, the entire expression \(729q^3 + 1331\) can be seen as \((9q)^3 + 11^3\). Recognizing these cubes helps us apply the sum of cubes formula efficiently. It is crucial to spot these forms because they allow us to use predefined algebraic identities. This helps in simplifying complex polynomials into manageable factors.

Polynomial Expressions

Polynomials are mathematical expressions consisting of variables and coefficients. They involve operations of addition, subtraction, multiplication, and non-negative integer exponents. In our example, the polynomial is \(729 q^3 + 1331\). Breaking it down:

• "729 \(q^3\)" is a term where 729 is a coefficient, \(q\) is a variable, and the exponent is 3.
• "1331" is a constant term, with no variable multiplying it.

Understanding the structure of polynomial expressions is key in recognizing forms like cubes. Polynomials can be manipulated using various algebraic methods to simplify, factor, or find roots. Recognizing these forms helps us decide which algebraic identities or formulas to apply, as shown in how we treated the initial polynomial problem. Keep in mind that dealing with polynomial expressions is about seeing these patterns.

Algebraic Identities

Algebraic identities are equalities that involve variables and hold true for any values of these variables. They are incredibly helpful in simplifying algebraic expressions. For problems involving cubes, we often use the formula for the **sum of cubes**: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).
In our exercise:

• Identify \(a = 9q\) and \(b = 11\) from the expression.
• Plug these into the sum of cubes formula.

Substitute and simplify as shown:

• First, expand \((9q)^2\) to get \(81q^2\).
• Then, calculate \((9q) \times 11\) resulting in \(99q\).
• Finally, square 11 to obtain \(121\).

The simplified expression is \( (9q + 11)(81q^2 - 99q + 121) \).
Using **algebraic identities** streamlines solving complex polynomial problems. Always remember that these identities are powerful tools in algebra, providing a means to break down and understand expressions better.