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Chapter 4: Problem 20

Divide. $$ \frac{q^{2}+2 q-35}{q-5} $$

Short Answer

Expert verified
\( q + 7 \)

Step by step solution

01

Factor the Numerator

Determine the factors of the numerator. The quadratic expression \( q^2 + 2q - 35 \) can be factored by finding two numbers that multiply to -35 and add up to 2. These numbers are 7 and -5. Thus, the factored form is: \( (q + 7)(q - 5) \).
02

Rewrite the Expression

Now, rewrite the original division problem by substituting the factored form of the numerator: \( \frac{(q + 7)(q - 5)}{q - 5} \).
03

Simplify the Fraction

Cancel out the common factor in the numerator and the denominator: \( \frac{(q + 7)\cancel{(q - 5)}}{\cancel{(q - 5)}} \). This leaves: \( q + 7 \).

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Key Concepts

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

Factoring Quadratic Expressions
Before simplifying a polynomial expression, we often need to factor it first. Factoring helps in breaking down complex expressions into simpler components. For the quadratic expression \( q^2 + 2q - 35 \), we find two numbers that multiply to \(-35\) (the constant term) and add up to \(2\) (the coefficient of the linear term). These two numbers are \(7\) and \(-5\). Therefore, we can write the expression as \( (q + 7)(q - 5) \). Factoring is a crucial step in polynomial division and helps in simplifying because it reveals common factors that may cancel out later.
Simplifying Rational Expressions
Once you've factored the numerator in a polynomial division, you rewrite the rational expression. For our example, the original expression \( \frac{q^2 + 2q - 35}{q - 5} \) becomes \( \frac{(q + 7)(q - 5)}{q - 5} \). Simplifying rational expressions involves canceling out common factors in the numerator and denominator. By expressing both parts in their simplest form, you can see that \( q - 5 \) is a common factor. This sets the stage for simplification by cancellation.
Canceling Common Factors
To simplify the expression further, we cancel the common factors in the numerator and the denominator. For \( \frac{(q + 7)(q - 5)}{q - 5} \), \( q - 5 \) appears in both the numerator and the denominator. Canceling these out leaves us with \( q + 7 \). This step is key in simplifying rational expressions and is only possible because we factored the quadratic expression first. Always ensure the expressions are fully factored before canceling to avoid errors.

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