Problem 89 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (x-3)(x-8) $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \((x-3)(x-8)\). Answer: The simplified algebraic expression is \(x^2 - 11x + 24\).

Step 1: Distribute \((x-3)\) to each term in \((x-8)\)

To distribute \((x-3)\) to \((x-8)\), we need to multiply \((x-3)\) to each term in the second parentheses, which gives: $$(x-3)(x) + (x-3)(-8)$$

Step 2: Distribute the terms further

Now we can distribute the terms in each of the expressions: $$x(x-3) - 8(x-3)$$ We can now distribute further: $$x^2 - 3x - 8x + 24$$

Step 3: Combine like terms

There are two like terms (-3x and -8x) that we can combine: $$x^2 - 11x + 24$$ The simplified algebraic expression is: $$x^2 - 11x + 24$$

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps you multiply a single term across terms that are inside parentheses. Think of it as a way to "distribute" multiplication over addition or subtraction within an expression. In our problem, we apply the distributive property to the expression \( (x-3)(x-8) \).
To use the distributive property here, we follow a few simple steps:

• Multiply each term within the first set of parenthesis \( (x-3) \) by each term in the second set of parenthesis \( (x-8) \).
• For instance, first multiply \( x \) by both terms in \( (x-8) \), and then multiply \(-3) \) by both terms in \( (x-8) \).
• This gives: \(x(x) - 3(x) + x(-8) - 3(-8) \).

Applying the distributive property can simplify complex algebraic expressions and is particularly useful in polynomial multiplication. It's a tool that helps ensure you don't miss any terms when multiplying.

Combining Like Terms

When simplifying algebraic expressions, combining like terms is an essential skill. Like terms are terms that have the same variable and exponent. In our expression \(x^2 - 3x - 8x + 24\), we notice that there are terms that can be combined for simplification.

• Identify terms that have identical variable factors. Here, \(-3x\) and \(-8x\) are like terms because both have the variable 'x' and the same exponent.
• Combine these terms by adding or subtracting their coefficients. So, \-3x - 8x = -11x\.
• This simplification results in a cleaner and simpler expression: \(x^2 - 11x + 24\).

By combining like terms, you reduce the complexity of the expression, making it easier to work with and understand. It's like tidying up to get a clearer picture!

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials to simplify the expression and is assisted greatly by the distributive property. In our example, the polynomials \( (x-3) \) and \( (x-8) \) need to be multiplied.

• We first apply the distributive property, multiplying each term in the first polynomial by each term in the second, resulting in individual terms like \( x(x)\), \(-3(x)\), \(x(-8)\), and \(-3(-8)\).
• This multiplication produces a series of products: \(x^2\), \(-3x\), \(-8x\), and \(24\).
• Once this is done, combining like terms simplifies the expression. We added \(-3x\) and \(-8x\) to get \(-11x\).

Understanding polynomial multiplication is crucial for solving a wide range of algebraic problems and goes hand in hand with other algebra skills like using the distributive property and combining like terms.