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Chapter 5: Problem 67

Multiply. $$ (a+b-c)(a+b+c) $$

Short Answer

Expert verified
The solution is: \( a^2 + 2ab + b^2 - c^2 \)

Step by step solution

01

Write down the problem

We need to multiply the expressions \[ (a + b - c)(a + b + c) \]
02

Apply the difference of squares

Notice that the expression is in the form of \[ (x - y)(x + y) \] where \( x = a + b \) and \( y = c \). This allows us to use the difference of squares formula: \[ (x - y)(x + y) = x^2 - y^2 \]
03

Substitute back the values

Substitute \( a + b \) for \( x \) and \( c \) for \( y \): \[ x = a + b \], \( y = c \). So: \[ (a+b-c)(a+b+c) = (a+b)^2 - c^2 \]
04

Expand the square

Expand \( (a + b)^2 \) using the binomial expansion formula: \[ (a + b)^2 = a^2 + 2ab + b^2 \]. Therefore, \[ (a+b)^2 - c^2 = a^2 + 2ab + b^2 - c^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

difference of squares
The 'difference of squares' is a powerful algebraic tool. It allows us to simplify products of binomials into simpler forms. The main formula to remember is \[ (x - y)(x + y) = x^2 - y^2 \].
This formula works because the product involves terms that cancel each other out, leading to a difference (subtraction) of two squares.
In this exercise, we recognize that \[ (a + b - c)(a + b + c) \] can be rewritten using this formula with:
  • \( x = a + b \)
  • \( y = c \)

Plugging these into the formula, we get \[ (a+b)^2 - c^2 \].
Recognizing this pattern is key and simplifies the multiplication process significantly.
binomial expansion
Binomial expansion helps us expand expressions that are squared, like \[ (a + b)^2 \].
The formula for any squared binomial is:
\[ (x + y)^2 = x^2 + 2xy + y^2 \].
In our exercise, we apply this to \[ (a + b)^2 \] and get:
\[ a^2 + 2ab + b^2 \].
This breaks down a more complex expression into simpler parts that are easier to handle.
Remembering and applying binomial expansion can greatly simplify many algebraic multiplication problems.
multiplying expressions
Multiplying algebraic expressions involves combining terms through distributive property or special formulas. For example, in the given problem:
\[ (a+b-c)(a+b+c) \],
we first identify if a special pattern like 'difference of squares' applies.
If it does, as in this case, we can use the formula directly:
\[ (a + b)^2 - c^2 \].
If a special pattern is not immediately apparent, use distributive properties to ensure every term in the first expression multiplies with every term in the second.
Using these strategies simplifies complex multiplication, making it more manageable and less error-prone.

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