Problem 87
Question
Simplify each rational expression. $$ \frac{x^{2}-8 x+16}{2 x-8} $$
Step-by-Step Solution
Verified Answer
\( \frac{x-4}{2} \) for \( x \neq 4 \).
1Step 1: Identify the Numerator and Denominator
The given rational expression is \( \frac{x^2 - 8x + 16}{2x - 8} \). The numerator is \( x^2 - 8x + 16 \), and the denominator is \( 2x - 8 \).
2Step 2: Factor the Numerator
The numerator \( x^2 - 8x + 16 \) is a quadratic trinomial. We look for two numbers that multiply to 16 and add up to -8. These numbers are -4 and -4, so we can factor the numerator as \( (x-4)(x-4) \) or \( (x-4)^2 \).
3Step 3: Factor the Denominator
The denominator \( 2x - 8 \) is a binomial. Notice that both terms have a common factor of 2, so we can factor it as \( 2(x - 4) \).
4Step 4: Simplify the Expression
Now we rewrite the rational expression with the factored numerator and denominator: \( \frac{(x-4)^2}{2(x-4)} \). Since \( x-4 \) is a common factor in both the numerator and denominator, we can cancel one \( x-4 \) from the top and bottom, resulting in \( \frac{x-4}{2} \).
5Step 5: Verify Simplification
Check that the simplified expression \( \frac{x-4}{2} \) is valid for every \( x \) except where the original denominator is zero. The original denominator, \( 2x-8=0 \), is zero when \( x=4 \). Thus, \( x eq 4 \) in the simplified expression.
Key Concepts
Factoring QuadraticsPolynomial DivisionCommon Factors
Factoring Quadratics
Factoring quadratics is a key skill in algebra, particularly when dealing with expressions like polynomials. In the exercise, we have a quadratic expression in the numerator, specifically \( x^2 - 8x + 16 \). This is known as a quadratic trinomial because it is a polynomial of degree 2 with three terms. The goal of factoring is to write this expression as a product of two binomials.
For a quadratic trinomial in the form \( ax^2 + bx + c \), you look for two numbers that multiply to \( c \) (the constant term) and add to \( b \) (the linear coefficient). In our case, we are searching for numbers that multiply to 16 and add to -8. Those numbers are -4 and -4. Therefore, the expression \( x^2 - 8x + 16 \) can be factored as \( (x - 4)(x - 4) \), which can also be written as \( (x - 4)^2 \).
Knowing how to factor quadratics will greatly aid in simplifying rational expressions because it reveals common factors that can be canceled between the numerator and denominator.
For a quadratic trinomial in the form \( ax^2 + bx + c \), you look for two numbers that multiply to \( c \) (the constant term) and add to \( b \) (the linear coefficient). In our case, we are searching for numbers that multiply to 16 and add to -8. Those numbers are -4 and -4. Therefore, the expression \( x^2 - 8x + 16 \) can be factored as \( (x - 4)(x - 4) \), which can also be written as \( (x - 4)^2 \).
Knowing how to factor quadratics will greatly aid in simplifying rational expressions because it reveals common factors that can be canceled between the numerator and denominator.
Polynomial Division
Polynomial division is another vital process in simplifying rational expressions. This exercise involves dividing the numerator by the denominator. However, before doing any division, we try to factor out common elements.
In our example, after factoring the quadratic numerator as \( (x-4)^2 \), we move on to the denominator \( 2x - 8 \). Although not strictly a polynomial division in the traditional long division sense, understanding division in terms of factoring helps us simplify expressions effectively.
Once everything is factored, and you have expressions like \( \frac{(x-4)^2}{2(x-4)} \), the division here is conceptual. We divide the common factor \( x-4 \) once from the numerator and once from the denominator, simplifying the expression to \( \frac{x-4}{2} \). The division is complete when common factors are canceled out where possible, simplifying the rational expression down to its simplest form.
In our example, after factoring the quadratic numerator as \( (x-4)^2 \), we move on to the denominator \( 2x - 8 \). Although not strictly a polynomial division in the traditional long division sense, understanding division in terms of factoring helps us simplify expressions effectively.
Once everything is factored, and you have expressions like \( \frac{(x-4)^2}{2(x-4)} \), the division here is conceptual. We divide the common factor \( x-4 \) once from the numerator and once from the denominator, simplifying the expression to \( \frac{x-4}{2} \). The division is complete when common factors are canceled out where possible, simplifying the rational expression down to its simplest form.
Common Factors
Identifying common factors in rational expressions is critical for simplification. After factoring both the numerator and the denominator, you look for expressions that appear in both. These are the common factors that can be canceled.
In the given exercise, the common factor is \( x-4 \). After factoring, the expression \( \frac{(x-4)^2}{2(x-4)} \) shows \( x-4 \) appearing in both the numerator and the denominator. Simplifying involves canceling out one of these \( x-4 \) terms in each part of the fraction, resulting in the simplified form \( \frac{x-4}{2} \).
It's important to ensure that canceling out common factors does not result in division by zero in the original expression. Thus, you always check where the original denominator equals zero (here, when \( x = 4 \)), which helps define excluded values, preserving the initial expression's constraints.
In the given exercise, the common factor is \( x-4 \). After factoring, the expression \( \frac{(x-4)^2}{2(x-4)} \) shows \( x-4 \) appearing in both the numerator and the denominator. Simplifying involves canceling out one of these \( x-4 \) terms in each part of the fraction, resulting in the simplified form \( \frac{x-4}{2} \).
It's important to ensure that canceling out common factors does not result in division by zero in the original expression. Thus, you always check where the original denominator equals zero (here, when \( x = 4 \)), which helps define excluded values, preserving the initial expression's constraints.
Other exercises in this chapter
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