Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. \(x^{3}+64 y^{3}=(\quad)^{3}+(\quad)^{3}\) b. \(8 x^{3}-27=(\quad)^{3}-(\quad)^{3}\)

Step-by-Step Solution

Answer

a. \((x)^3 + (4y)^3\), b. \((2x)^3 - (3)^3\)

Step 1: Recognize Cube Sum Identity

The expression \(x^3 + 64y^3\) follows the sum of cubes formula, \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b^3 = 64y^3\).

Step 2: Identify Cube Roots

Identify the cube root of each term: \(x^3\) yields \(x\) and \((64y^3)^{1/3} = (4y)\). Thus, \((x)^3 + (4y)^3\).

Step 3: Verify the Expression

Check the expression: \(x^3 + 64y^3 = (x)^3 + (4y)^3\). Thus, the missing terms for the blanks are \(x\) and \(4y\).

Step 4: Recognize Cube Difference Identity

The expression \(8x^3 - 27\) follows the difference of cubes formula, \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a^3 = 8x^3\) and \(b^3 = 27\).

Step 5: Identify Cube Roots in the Difference

Identify the cube root of each term: \((8x^3)^{1/3} = (2x)\) and \((27)^{1/3} = 3\). Thus, \((2x)^3 - (3)^3\).

Step 6: Verify the Expression for the Difference

Check the expression: \(8x^3 - 27 = (2x)^3 - (3)^3\). Thus, the missing terms for the blanks are \(2x\) and \(3\).

Key Concepts

Sum of Cubes

The sum of cubes is a fascinating algebraic identity where you can break down an expression in the form of the sum of two cubes. The formula for the sum of cubes is \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).

For example, let's look at an expression like \(x^3 + 64y^3\). Here, it's crucial to recognize that each component is a perfect cube. Specifically, \(x^3\) is the cube of \(x\), and \(64y^3\) can be rewritten as \((4y)^3\).

To apply the sum of cubes formula, set \(a = x\) and \(b = 4y\). Then, substitute these into the formula to expand it:

• First, add the cube roots \((a + b)\), which gives \(x + 4y\).
• Then, calculate \(a^2 - ab + b^2\), breaking these down as \(x^2 - x\cdot4y + (4y)^2\).
• Combine these terms to verify they match the original \(a^3 + b^3\).

This identity allows you to neatly factorize a sum of cubes.

Difference of Cubes

Difference of cubes is another essential algebraic identity that lets us factor expressions like \(a^3 - b^3\). The formula is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).

Looking at \(8x^3 - 27\), we can recognize it as a difference of cubes. Here, \(8x^3\) can be rewritten as \((2x)^3\) and \(27\) as \(3^3\).

Once again, identify \(a\) and \(b\):

• For \((2x)^3\), \(a = 2x\).
• For \((3)^3\), \(b = 3\).

Substitute these into the difference of cubes formula as follows:

• Subtract the cube roots \((a-b)\) to get \((2x - 3)\).
• Then expand \(a^2 + ab + b^2\) as \((2x)^2 + 2x\cdot3 + 3^2\).

Confirming the identity ensures that the expression correctly factors the difference of the original cubes.

Cube Roots

Cube roots are numbers that, when multiplied by themselves three times, result in the original number. Understanding cube roots helps in simplifying cubed expressions to identify underlying algebraic formulas.

For instance, to resolve expressions like \(x^3\) or \(64y^3\), we need their cube roots:

• The cube root of \(x^3\) is just \(x\).
• For \(64y^3\), observe \(64 = 4^3\) and \(y^3\) is inherent, making the cube root \(4y\).

Similarly, consider the expression \(8x^3 - 27\):

• The cube roots would be \((2x)\) because \(8 = 2^3\), and \(x^3\) offers \(x\).
• And \(3\) for \((27)\) since \(27 = 3^3\).

Solving cube roots bridges the gap to applying algebraic identities effectively.

Factoring

Factoring is the process of breaking down an algebraic expression into simpler components that multiply together to give the original expression. It's like solving a puzzle where you find pieces that fit together neatly.

Factoring is crucial when dealing with algebraic identities such as the sum or difference of cubes. For instance, once you've identified the cube roots in expressions such as \(x^3 + 64y^3\) or \(8x^3 - 27\), you can use factoring to verify their forms and make them easier for further calculations.

In practice, perform these steps to factor:

• Recognize each component—whether an expression is a sum or difference of cubes.
• Apply the correct formula \((a+b)(a^2 - ab + b^2)\) for sums and \((a-b)(a^2 + ab + b^2)\) for differences.
• Break down each part to match the patterns the formula provides.

By applying these steps, you can transform complicated polynomial equations into products of simpler terms, making problems more approachable for solving or further analysis.