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Chapter 7: Problem 55

Multiply and simplify each of the following. Whenever possible, do the multiplication of two binomials mentally. $$(x+7)(x+7)$$

Short Answer

Expert verified
The simplified form is \( x^2 + 14x + 49 \).

Step by step solution

01

Recognize the Binomial

The expression \((x+7)(x+7)\) can be recognized as a binomial multiplied by itself.
02

Apply the Binomial Square Formula

Use the formula \[ (a + b)^2 = a^2 + 2ab + b^2 \] to simplify the expression \( (x+7)^2 \). Here, \(a = x \) and \(b = 7 \).
03

Calculate Each Term

Calculate \( a^2 \), \( 2ab \), and \( b^2 \):\[ a^2 = x^2 \] \[ 2ab = 2 \times x \times 7 = 14x \] \[ b^2 = 7^2 = 49 \]
04

Combine the Terms

Combine all the terms to obtain the simplified expression: \[ x^2 + 14x + 49 \]

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Key Concepts

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

binomial square formula
When you multiply a binomial by itself, it is called squaring the binomial. The binomial square formula helps us do this efficiently:
\( (a + b)^2 \).
The formula states that \[ (a + b)^2 = a^2 + 2ab + b^2 \]. This formula is a simpler way to handle polynomial expansion of a binomial.
For our example, we have \( (x+7)^2 \). Here, \( a = x \) and \( b = 7 \).
Applying the binomial square formula:
  • Find \( a^2 \)
  • Multiply \( 2 \) by both \( a \) and \( b \)
  • Find \( b^2 \)
This results in the expanded form: \( x^2 + 14x + 49 \).
algebra simplification
Simplification in algebra means making the expression more manageable or easier to read.
After expanding a polynomial, combining like terms further simplifies the expression.
Let's break down the simplification after polynomial expansion:
  • First, expand the binomial using the binomial square formula.
  • Combine all like terms (terms with the same variables and powers).
  • Ensure there are no further simplifications possible.
In our example \( (x+7)(x+7) \):
Expansion gives \( x^2 + 14x + 49 \).
There are no like terms to combine, so the simplified expression remains \( x^2 + 14x + 49 \).
polynomial expansion
Polynomial expansion involves multiplying expressions and applying distributive properties.
For binomials, we often use formulas like \[ (a + b)^2 = a^2 + 2ab + b^2 \].
It distributes each term in the first polynomial to every term in the second polynomial.
Given \( (x+7)(x+7) \): Start by applying the formula:
\[ (x+7)^2 = x^2 + 2(7)(x) + 7^2 \]. This becomes:
\( x^2 + 14x + 49 \).
Always check each step in the expansion to ensure accuracy. Double-check the operations to make sure they follow the rules of algebra simplification.

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