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Chapter 5: Problem 69

\(\left(c^{2}+8\right) \div(c+6)\)

Short Answer

Expert verified
\(\frac{c^2 + 8}{c + 6}\) is already in simplest form.

Step by step solution

01

- Simplify the Expression

Look at the numerator and the denominator of the given algebraic fraction. The fraction is \(\frac{c^2 + 8}{c + 6}\). Check whether the numerator can be factored for any common factors.
02

- Check for Factorization

Check if \( c^2 + 8 \) can be factored. In this case, \( c^2 + 8 \) cannot be factored further as it does not take the form of any known factorable polynomials like difference of squares, perfect square trinomials, or others.
03

- Division

Since the numerator \(c^2 + 8\) does not factor and does not simplify with \(c + 6\), the expression remains as it is. Thus, \(\frac{c^2 + 8}{c+6}\) is already in its simplest form.

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Key Concepts

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

numerator and denominator
In algebraic fractions, understanding the numerator and denominator is crucial. The numerator is the top part of the fraction, while the denominator is the bottom part. In the given exercise, our fraction is \(\frac{c^2 + 8}{c + 6}\). Here,
  • The numerator is \(c^2 + 8\)
  • The denominator is \(c + 6\)

To simplify an algebraic fraction, you need to understand both parts separately. Always start by examining these components.
factorization
Factorization involves breaking down complex expressions into simpler 'factors' that you can multiply to get the original expression. For our exercise, we need to see if the numerator \(c^2 + 8\) can be factorized.
Known factorization techniques include:
  • Factoring out common terms
  • Factoring quadratics by splitting the middle term
  • Using special identities like the difference of squares

For \(c^2 + 8\), there are no common factors, nor does it fit any special factorization forms, meaning it cannot be factorized further.
simplifying expressions
Simplifying algebraic expressions means making them as simple as possible. It involves canceling common factors in the numerator and denominator, reducing fractions, or combining like terms.

For the given fraction \(\frac{c^2 + 8}{c + 6}\), we first checked for factorization, but since it wasn't possible, we confirmed that the fraction is already in its simplest form. Here are general steps to simplify:
  • Check for common factors in numerator and denominator.
  • Factorize if possible, and then cancel the common factors.
  • If no further simplification is possible, the fraction is in its simplest form.

This ensures that you have the most manageable version of the expression.

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Most popular questions from this chapter

The base of a triangle is 4 in. longer than twice the height. a. If \(h=\) height, write a polynomial expression in \(h\) that represents the base, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(h\) that represents the area.

\(\left(x^{2}-25\right) \div(x+5)\)

A drafter is making enlargements of a rectangular drawing that preserve the relative width and length of the drawing. The length of the drawing is five- fourths of the width. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

\(\left(k^{7}-5 k^{3}+100 k+20\right) \div 5\)

a. Simplify: \((x-5)(x+8)\) b. Simplify: \((x+5)(x-8)\) c. Describe the difference in the products.
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