Problem 75 - Algebra 2

Question

Find each product. $$ (a+b)(a+2 b) $$

Step-by-Step Solution

Answer

The product is \(a^2 + 3ab + 2b^2\).

Step 1: Apply the Distributive Property

To find the product of two binomials, we need to apply the distributive property. Distribute each term in the first binomial across the second binomial: \[(a+b)(a+2b) = a(a+2b) + b(a+2b)\]

Step 2: Distribute Each Term Separately

Now distribute each term separately within the parentheses: * Distribute \(a\): \[a(a+2b) = a \cdot a + a \cdot 2b = a^2 + 2ab\]* Distribute \(b\): \[b(a+2b) = b \cdot a + b \cdot 2b = ab + 2b^2\]

Step 3: Combine Like Terms

Next, combine the like terms from the expression obtained in the previous step:\[a^2 + 2ab + ab + 2b^2\] Add the like terms \(2ab\) and \(ab\): \[a^2 + 3ab + 2b^2\]

Step 4: Write the Final Expression

After combining like terms, the final expression of the product is: \[a^2 + 3ab + 2b^2\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that lets us multiply a single term by two or more terms inside a set of parentheses. In the problem \((a+b)(a+2b)\), the distributive property allows us to tackle each part of the multiplication separately.
By distributing each term of the first binomial across the second binomial, we transform the expression into something more manageable.

• First, multiply both terms inside the second binomial,\((a+2b)\), by the first term of the first binomial, \(a\).
• Then, do the same for the second term of the first binomial, \(b\).

This method breaks down the expression into simpler parts we can easily handle, moving to the next steps to solve the equation efficiently.

Combining Like Terms

Combining like terms is an essential skill to simplify expressions by merging terms with the same variable and power.In our solution, after applying the distributive property, we arrive at the expanded expression: \[a^2 + 2ab + ab + 2b^2\].
"Like terms" are terms that have identical variable components. In the expression, \(2ab\) and \(ab\) are like terms because they both contain \(a\) and \(b\) to the first power.

• We add these coefficients to simplify the expression, resulting in \(3ab\).
• The operation simplifies our polynomial to \(a^2 + 3ab + 2b^2\).

Combining like terms reduces unnecessary complexity in expressions and often provides a more readable and workable form.

Polynomial Expression

A polynomial expression consists of multiple terms combined by addition or subtraction, where each term contains variables raised to non-negative integer powers. In the example \[(a+b)(a+2b)\],when expanded using distributive property and combining like terms, we obtain \[a^2 + 3ab + 2b^2\].
It's vital to recognize different parts of a polynomial, such as:

• "Terms" like \(a^2\), \(3ab\), and \(2b^2\).
• "Coefficients," which are the numerical parts of the terms, like 1 in \(a^2\) and 3 in \(3ab\).

Understanding each part gives insight into manipulating and solving polynomial equations, a skill necessary in algebra and higher math.

Algebraic Manipulation

Algebraic manipulation refers to methods used to rearrange and simplify expressions in algebra. In solving \((a+b)(a+2b)\), we applied various manipulation techniques:

• We used the distributive property to expand the product of two binomials.
• Combining like terms to further simplify the result.

These steps: distribution and combining, transform complex expressions into more straightforward, equivalent forms.
Such manipulations are critical in solving algebraic equations, enabling the rounding down of long and convoluted expressions into simple, understandable equations.Whether it's balancing equations or factoring, these techniques underpin algebraic problem-solving.