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Chapter 5: Problem 90

Compute and simplify. $$ (x+5)^{2}-(x-3)^{2} $$

Short Answer

Expert verified
16(x + 1)

Step by step solution

01

Apply the Binomial Theorem

Expand each square term using the binomial theorem. For \((x+5)^2\), we have \((x+5)^2 = x^2 + 2x \cdot 5 + 5^2 = x^2 + 10x + 25\). For \((x-3)^2\), we have \((x-3)^2 = x^2 - 2x \cdot 3 + 3^2 = x^2 - 6x + 9\).
02

Substitute Expanded Binomials Back

Insert the expanded forms back into the original expression: \[(x+5)^2 - (x-3)^2 = (x^2 + 10x + 25) - (x^2 - 6x + 9)\]
03

Distribute the Negative Sign

Distribute the negative sign through the second polynomial: \[(x^2 + 10x + 25) - x^2 + 6x - 9\]
04

Combine Like Terms

Combine like terms to simplify the expression: \[x^2 + 10x + 25 - x^2 + 6x - 9 = (x^2 - x^2) + (10x + 6x) + (25 - 9) = 16x + 16\]
05

Factor Out Common Terms

Factor out the greatest common factor from the resulting expression: \[16x + 16 = 16(x + 1)\]

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Key Concepts

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

Expanding Binomials
When you see a binomial squared, like \( (x + 5)^2 \), it means you're multiplying the binomial by itself. Expanding binomials involves applying the Binomial Theorem, which helps break down the equation. For example, \( (x+5)^2 \) expands to \( x^2 + 2x \cdot 5 + 5^2 \). This yields \( x^2 + 10x + 25 \). Similarly, \( (x-3)^2 \) expands to \( x^2 - 2x \cdot 3 + 3^2 \), giving us \( x^2 - 6x + 9 \). Expanding binomials is fundamental for simplifying later steps.
Combining Like Terms
Combining like terms helps to simplify polynomials. Look for terms with the same variable and exponent. In our example, substituting the expanded binomials back into \( (x+5)^2 - (x-3)^2\), we get:
  • \((x^2 + 10x + 25) - (x^2 - 6x + 9)\)
Next, we distribute the negative sign through the second polynomial:
  • \((x^2 + 10x + 25) - x^2 + 6x - 9\)
Now, identify and combine the like terms:
  • \( x^2 \) terms: \( x^2 - x^2 \)
  • \( x \) terms: \( 10x + 6x \)
  • Constant terms: \( 25 - 9 \)
Adding these together gives us 16\( x + 16\).
Factoring Polynomials
Factoring polynomials involves extracting the greatest common factor (GCF) from the expression. In our example, the simplified expression is \(16x + 16\). Here, both terms share a GCF of 16. So, we factor out the 16: \(16(x + 1)\). This last step simplifies the expression and makes it easier to understand and use in further calculations.
  • Find the GCF of all terms
  • Extract the GCF and place the remaining polynomial inside the parenthesis
This step-by-step process ensures the polynomial is expressed in its simplest form.

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