Problem 89 - Beginning and Intermediate Algebra

Question

Find the following special products. $$\left(\frac{2}{3}-k\right)\left(\frac{2}{3}+k\right)$$

Step-by-Step Solution

Answer

The special product for the given expressions is: \(\left(\frac{2}{3}-k\right)\left(\frac{2}{3}+k\right) = \frac{4}{9} - k^2\)

Step 1: Identify A and B

In the given expression, we have: A = \(\frac{2}{3}\) and B = k

Step 2: Apply the difference of squares formula

The difference of squares formula is: (A - B)(A + B) = A^2 - B^2. Plug in A = \(\frac{2}{3}\) and B = k into this formula: \(\left(\frac{2}{3}-k\right)\left(\frac{2}{3}+k\right) = \left(\frac{2}{3}\right)^2 - k^2\)

Step 3: Calculate A^2 and B^2

Calculate the square of A and B, which are \(\frac{2}{3}\) and k respectively. \(A^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}\) and \(B^2 = k^2\)

Step 4: Substitute A^2 and B^2 in the formula

Now substitute the values of A^2 and B^2 back into the difference of squares formula: \(\left(\frac{2}{3}-k\right)\left(\frac{2}{3}+k\right) = \frac{4}{9} - k^2\) So, the special product for the given expressions is: \(\left(\frac{2}{3}-k\right)\left(\frac{2}{3}+k\right) = \frac{4}{9} - k^2\)

Key Concepts

Special Products

Special products are specific algebraic expressions that result from multiplying simple polynomials and have recognizable forms. One such product is the difference of squares, which arises from the product of a sum and a difference of two terms.
Understanding special products like the difference of squares is essential because they allow us to simplify expressions quickly without multiplying every term. This occurs in expressions like \((a-b)(a+b) = a^2 - b^2\).
There are several other well-known special products, including:

• Perfect square trinomials: \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\).
• Difference of squares: \((a-b)(a+b) = a^2 - b^2\).

Recognizing these forms helps in factoring solutions quickly and efficiently, streamlining the process of solving algebraic equations and simplifying expressions.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations to convey mathematical relationships. They are central components in algebra and expressing complex relationships in a simplified manner.
For instance, \(x^2 + 3x + 2\) is an algebraic expression showing how terms and operations are organized around the variable \(x\).
Understanding how to manipulate these expressions using rules of operations allows us to make sense of mathematical problems. Some core features of algebraic expressions include:

• Terms: The separate parts of an expression, such as \(x\) or \(5\).
• Coefficients: The numerical part that multiplies a variable, e.g., in \(3x\), 3 is the coefficient.
• Variables: Symbols that represent unknown values, commonly \(x, y, z\) in expressions.

While simplifying or manipulating algebraic expressions, it is vital to apply the correct operations and maintain the balance of equations. Mastery of algebraic expressions leads to proficiency in solving equations and understanding mathematical concepts.

Multiplication of Binomials

Multiplying binomials involves using the distributive property to simplify expressions. Binomials are algebraic expressions with two terms, such as \((x+2)(x-3)\), and their multiplication often leads to polynomials.
The process of multiplying binomials is often remembered through the acronym FOIL:

• First: Multiply the first terms from each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

Applying these steps systematically helps break down the multiplication into manageable parts, ensuring no terms are missed. For example, when multiplying \((x+2)(x-3)\):\[x \times x = x^2\] \[x \times (-3) = -3x\] \[2 \times x = 2x\] \[2 \times (-3) = -6\]Combining these results gives \(x^2 - 3x + 2x - 6 = x^2 - x - 6\).
This method emphasizes the meticulous organization of terms and is fundamental in simplifying and working with larger algebraic expressions.