Problem 37
Question
For the following exercises, factor the polynomials. $$ x^{3}+216 $$
Step-by-Step Solution
Verified Answer
\((x + 6)(x^2 - 6x + 36)\)
1Step 1: Identify the Type of Polynomial
The expression \( x^3 + 216 \) is a sum of cubes. The sum or difference of cubes can be factored using specific formulas. Here, we recognize that \(216\) is a perfect cube, since \( 216 = 6^3 \).
2Step 2: Use the Sum of Cubes Formula
For the sum of cubes \( a^3 + b^3 \), the factorization formula is \( (a + b)(a^2 - ab + b^2) \). Here, \( a = x \) and \( b = 6 \).
3Step 3: Substitute into the Formula
Substitute \( a = x \) and \( b = 6 \) into the sum of cubes formula: \[(x + 6)(x^2 - x \cdot 6 + 6^2) \].
4Step 4: Simplify the Expression
Simplify the expression from Step 3. The second term becomes:\[ x^2 - 6x + 36 \].So, the factored form of the polynomial \( x^3 + 216 \) is \( (x + 6)(x^2 - 6x + 36) \).
Key Concepts
Sum of CubesPerfect CubesFactored FormAlgebraic Expressions
Sum of Cubes
A sum of cubes is an algebraic expression that takes the form \( a^3 + b^3 \). In this situation, each term in the expression is a "cube" because it is raised to the third power, like \( x^3 \) and \( 216 \) in the example.
The sum of cubes can be effectively factored using a specific formula: \((a^3 + b^3 = (a + b)(a^2 - ab + b^2))\). By simply identifying the components \( a \) and \( b \), we can easily substitute them into this formula.
The sum of cubes can be effectively factored using a specific formula: \((a^3 + b^3 = (a + b)(a^2 - ab + b^2))\). By simply identifying the components \( a \) and \( b \), we can easily substitute them into this formula.
- The expression \( x^3 + 216 \) is a sum of cubes because \( 216 \) is a perfect cube of \( 6 \), or \( 6^3 \).
- Recognizing these cubes allows us to break the expression down using the sum of cubes formula, which simplifies complex algebraic expressions.
Perfect Cubes
Perfect cubes are numbers or expressions that can be written as the cube of an integer or another simpler term. A perfect cube has a straightforward mathematical property where it equals another number raised to the power of three.
In the original exercise, the number 216 is crucial because it recognizes as a perfect cube \(( 6^3 )\). This understanding facilitates utilizing the sum of cubes factorization formula as detailed previously.
In the original exercise, the number 216 is crucial because it recognizes as a perfect cube \(( 6^3 )\). This understanding facilitates utilizing the sum of cubes factorization formula as detailed previously.
- Identifying perfect cubes helps simplify polynomial expressions by reducing them into more manageable factors.
- For instance, knowing that 216 is \( 6^3 \) allows us to use the variables \( a = x \) and \( b = 6 \) in the factorization formula.
Factored Form
The factored form of a polynomial is a way of rewriting a polynomial as a product of simpler polynomials. This makes it easier to solve for the polynomial's roots and analyze its behavior.
For the sum of cubes expression \( x^3 + 216 \), the factored form is \((x + 6)(x^2 - 6x + 36)\). This form is derived using the sum of cubes factorization formula mentioned earlier.
For the sum of cubes expression \( x^3 + 216 \), the factored form is \((x + 6)(x^2 - 6x + 36)\). This form is derived using the sum of cubes factorization formula mentioned earlier.
- The factored form provides a clearer perspective of a polynomial's structure, revealing possible solutions or intercepts where the polynomial might equal zero.
- It is a critical step in simplifying expressions and solving equations.
Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, operators, and variables like \( x, a, \) or \( b \). These expressions don't contain an equality sign, differentiating them from equations.
They allow us to represent problems and real-world scenarios in a mathematical form, and functions like polynomial factorization help simplify or manipulate these expressions.
They allow us to represent problems and real-world scenarios in a mathematical form, and functions like polynomial factorization help simplify or manipulate these expressions.
- The expression \( x^3 + 216 \) is an algebraic expression and through factorization, is transformed into a more interpretable form \((x + 6)(x^2 - 6x + 36)\).
- Algebraic expressions are the building blocks of algebra and critical in learning more complex mathematical operations.
Other exercises in this chapter
Problem 36
For the following exercises, solve for the variable. $$ 4(3-1) x=4 $$
View solution Problem 36
Solve for the variable. $$ 4(3-1) x=4 $$
View solution Problem 37
For the following exercises, divide the rational expressions. $$ \frac{y+3}{y-2}+\frac{y-3}{y+1} $$
View solution Problem 37
For the following exercises, simplify each expression. $$ \sqrt{49 p} $$
View solution