Problem 51 - Elementary Algebra

Question

a. Simplify: \((x+5)(x+8)\) b. Simplify: \((x-5)(x-8)\) c. Describe the difference in the products.

Step-by-Step Solution

Answer

The products differ in the sign of the linear term: \( x^2 + 13x + 40 \) and \( x^2 - 13x + 40 \).

Step 1: Expand \( (x+5)(x+8) \)

Use the distributive property to expand the expression \( (x+5)(x+8) \). Multiply each term in the first parenthesis by each term in the second parenthesis: \[ (x+5)(x+8) = x \times x + x \times 8 + 5 \times x + 5 \times 8 \]

Step 2: Simplify \( (x+5)(x+8) \)

Combine the like terms from the expansion: \[ x^2 + 8x + 5x + 40 = x^2 + 13x + 40 \]

Step 3: Expand \( (x-5)(x-8) \)

Similarly, use the distributive property to expand \( (x-5)(x-8) \): \[ (x-5)(x-8) = x \times x + x \times (-8) + (-5) \times x + (-5) \times (-8) \]

Step 4: Simplify \( (x-5)(x-8) \)

Combine the like terms from the expansion: \[ x^2 - 8x - 5x + 40 = x^2 - 13x + 40 \]

Step 5: Compare the products

Observe the difference between the simplified forms of the two expressions: \[ (x+5)(x+8) = x^2 + 13x + 40 \] \[ (x-5)(x-8) = x^2 - 13x + 40 \] The difference lies in the sign of the linear term.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra. It states that multiplying a single term by two or more terms within a parenthesis is the same as multiplying each term inside the parenthesis by the single term outside. This is essential for polynomial multiplication. For instance, in the expression \( (x+5)(x+8) \), we apply the distributive property:

\[ (x+5)(x+8) = x \times x + x \times 8 + 5 \times x + 5 \times 8 \]

This means we distribute the multiplication over each term, ensuring all combinations of terms are considered. Similarly, for \( (x-5)(x-8) \), we do:

\[ (x-5)(x-8) = x \times x + x \times (-8) + (-5) \times x + (-5) \times (-8) \]

Understanding and applying the distributive property helps to systematically tackle polynomial multiplication, breaking down complex expressions into manageable components.

Combining Like Terms

After applying the distributive property, the next step is to simplify by combining like terms. Like terms are terms that have the same variable raised to the same power.

For example, after expanding \( (x+5)(x+8) \), we get:

\[ x^2 + 8x + 5x + 40 \]

We then combine the like terms \( 8x \) and \( 5x \) to simplify:

\[ x^2 + 13x + 40 \]

Similarly, for \( (x-5)(x-8) \), after expansion:

\[ x^2 - 8x - 5x + 40 \]

We combine \( -8x \) and \( -5x \) to get:

\[ x^2 - 13x + 40 \]

Combining like terms simplifies the expressions, making them easier to work with and understand.

Comparison of Expressions

After simplifying both expressions, \( (x+5)(x+8) \) and \( (x-5)(x-8) \), we observe:

\[ (x+5)(x+8) = x^2 + 13x + 40 \]

\[ (x-5)(x-8) = x^2 - 13x + 40 \]

Notice that both expressions have the same quadratic term \( x^2 \) and constant term \( 40 \). The key difference is in the linear term. In the first expression, the linear term is positive ( +13x ), while in the second expression, it is negative ( -13x ).

This comparison shows how the signs of the numbers in the binomials affect the resulting polynomial. The positive and negative signs change the direction and value of the linear term, even though other terms remain unaffected. Understanding this helps in predicting the nature of polynomials formed by different binomials.