Problem 65 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$(x+10)(a+b+c)$$

Step-by-Step Solution

Answer

Question: Expand the given expression \((x+10)(a+b+c)\) using the distributive property. Answer: \((x+10)(a+b+c) = xa + 10a + xb + 10b + xc + 10c\)

Step 1: Distribute \((x+10)\) to \(a\)

Using the distributive property, distribute \((x+10)\) to \(a\): $$(x+10)(a) = xa + 10a$$

Step 2: Distribute \((x+10)\) to \(b\)

Using the distributive property, distribute \((x+10)\) to \(b\): $$(x+10)(b) = xb + 10b$$

Step 3: Distribute \((x+10)\) to \(c\)

Using the distributive property, distribute \((x+10)\) to \(c\): $$(x+10)(c) = xc + 10c$$

Step 4: Combine the results

Combine the results from Steps 1, 2, and 3 to obtain the expanded expression: $$(x+10)(a+b+c) = xa + 10a + xb + 10b + xc + 10c$$

Key Concepts

Algebraic Expansion

Algebraic expansion is a fundamental concept in algebra. It is the process by which expressions are simplified by removing parentheses through multiplication. This expansion helps in simplifying complex expressions and making calculations easier.
When you apply the distributive property to expand an expression such as \((x+10)(a+b+c)\), you are essentially multiplying each term inside the parenthesis by each part outside the parenthesis.

• The term \((x+10)\) is multiplied separately with \(a\), \(b\), and \(c\).
• This step-by-step technique assures that each component is correctly accounted for in the final expression.

So, the procedure allows you to transform \((x+10)(a+b+c)\) into a more manageable algebraic form by methodically distributing each term, leading to a streamlined polynomial for easier handling in further calculations.

Polynomial Multiplication

Polynomial multiplication involves multiplying each part of one polynomial by every part of another. This method is applied using the distributive property, which is fundamental when dealing with expressions like \((x+10)(a+b+c)\).
For this particular expression:

• The polynomial \((x+10)\) is regarded as one binomial that interacts with another polynomial \((a+b+c)\).
• Each term inside one polynomial multiplies each term of the other polynomial. In simpler terms, multiply every part in \((x+10)\) by each part of \(a\), \(b\), and \(c\).

The process may seem lengthy, but it simplifies polynomial expressions and paves the way for solving more complex problems, being an essential skill to master in algebra.

Step-by-Step Algebra

Approaching algebraic problems step-by-step ensures clarity and a deeper understanding of the underlying principles.
In the solution of \((x+10)(a+b+c)\):

• Each term inside the parentheses is tackled one at a time. This begins by distributing the first polynomial term \((x+10)\) to each term inside the second polynomial \((a+b+c)\).
• For simplification, distribute to \(a\), followed by \(b\), and finally \(c\), ensuring no steps are skipped.
• This results in the expression: \(xa + 10a + xb + 10b + xc + 10c\).

Breaking down the complete process into smaller, manageable steps makes the algebraic expansion easier and amplifies your overall confidence in handling polynomials.