Simplify ((21x-147)/(7x-63))/((x^2-49)/(x^2-3x-54))
The given problem is asking to perform algebraic simplification of a complex rational expression. A rational expression is a fraction where both the numerator and the denominator are polynomials. In this case, the problem involves two rational expressions, one being divided by the other. To simplify the overall expression, you need to factorize the polynomials where possible and then look for common factors that can be cancelled out from the numerator and the denominator. Additionally, you will apply the rule that dividing by a fraction is equivalent to multiplying by its reciprocal. The goal is to simplify the expression to its most reduced form.
$\frac{\frac{21 x - 147}{7 x - 63}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$
Answer
Solution:
Step 1: Simplify the numerator and denominator
Step 1.1: Factor out common terms
Extract the factor of $7$ from $21x$ and $-147$.
$$\frac{\frac{7(3x) - 147}{7x - 63}}{\frac{x^2 - 49}{x^2 - 3x - 54}}$$
Step 1.2: Simplify the fraction
Factor out $7$ from both terms in the numerator.
$$\frac{\frac{7(3x - 21)}{7x - 63}}{\frac{x^2 - 49}{x^2 - 3x - 54}}$$
Step 1.3: Reduce common factors
Eliminate the common factor of $7$ in the numerator and denominator.
$$\frac{\frac{\cancel{7}(3x - 21)}{\cancel{7}(x - 9)}}{\frac{x^2 - 49}{x^2 - 3x - 54}}$$
Step 1.4: Finalize the numerator
Rewrite the simplified numerator.
$$\frac{3x - 21}{x - 9}$$
Step 2: Multiply by the reciprocal
Multiply the simplified numerator by the reciprocal of the denominator.
$$\frac{3x - 21}{x - 9} \cdot \frac{x^2 - 3x - 54}{x^2 - 49}$$
Step 3: Factor common terms in the numerator
Step 3.1: Extract the factor of $3$
Factor $3$ from $3x - 21$.
$$\frac{3(x - 7)}{x - 9} \cdot \frac{x^2 - 3x - 54}{x^2 - 49}$$
Step 4: Factor the quadratic expression
Step 4.1: Apply the AC method
Find two numbers that multiply to $-54$ and add to $-3$.
Step 4.2: Rewrite the quadratic
Express the quadratic as a product of its factors.
$$\frac{3(x - 7)}{x - 9} \cdot \frac{(x - 9)(x + 6)}{x^2 - 49}$$
Step 5: Simplify the denominator
Step 5.1: Express $49$ as a square
Rewrite $49$ as $7^2$.
Step 5.2: Factor using difference of squares
Factor the denominator as the difference of squares.
$$\frac{3(x - 7)}{x - 9} \cdot \frac{(x - 9)(x + 6)}{(x + 7)(x - 7)}$$
Step 6: Cancel common factors
Step 6.1: Eliminate $x - 7$
Remove the common factor $x - 7$.
$$\frac{3}{x - 9} \cdot \frac{(x - 9)(x + 6)}{x + 7}$$
Step 6.2: Eliminate $x - 9$
Remove the common factor $x - 9$.
$$3 \cdot \frac{x + 6}{x + 7}$$
Step 6.3: Combine terms
Combine the constant $3$ with the fraction.
$$\frac{3(x + 6)}{x + 7}$$
Knowledge Notes:
The problem-solving process involves simplifying a complex rational expression. The steps include factoring common terms, canceling like factors, and multiplying by the reciprocal of the denominator. Key knowledge points include:
Factoring expressions: Recognizing and factoring out common terms from polynomials.
Simplifying fractions: Reducing fractions by canceling common factors in the numerator and denominator.
Reciprocal multiplication: Multiplying a fraction by the reciprocal of another fraction effectively divides the fractions.
Difference of squares: Factoring expressions of the form $a^2 - b^2$ into $(a + b)(a - b)$.
AC method for factoring quadratics: Finding two numbers that multiply to the product of the quadratic's leading coefficient and constant term (AC) and add to the middle term's coefficient (B), then using these numbers to factor the quadratic.
These concepts are foundational in algebra and are used frequently in calculus, engineering, and the physical sciences. Understanding how to manipulate and simplify algebraic expressions is crucial for solving a wide range of mathematical problems.
