Simplify ((x^2-10x+24)/(x^2+x-42)*(x^2-49)/(x^2-11x+28))÷((3x^2-147)/(x^2-49))
This mathematics problem involves simplifying a complex rational expression. The rational expression consists of a combination of polynomial expressions both in the numerators and denominators. The task is to perform the operations of multiplication and division between these fractions while simplifying by factoring polynomials, cancelling out like terms, and reducing the expression to its simplest form. The final outcome should be a rational expression that is fully simplified, with no common factors in the numerator and denominator apart from 1.
$\frac{x^{2} - 10 x + 24}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2} - 11 x + 28} \div \frac{3 x^{2} - 147}{x^{2} - 49}$
Answer
Solution:
Simplifying the Expression
Step 1:
To divide fractions, multiply by the inverse of the divisor. Multiply $\frac{x^{2} - 10x + 24}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2} - 11x + 28}$ by the reciprocal of $\frac{3x^{2} - 147}{x^{2} - 49}$.
Step 2:
Factor the quadratic $x^{2} - 10x + 24$.
Find two numbers that multiply to $24$ and add to $-10$. These are $-6$ and $-4$.
Express the factored form as $(x - 6)(x - 4)$.
Step 3:
Factor the quadratic $x^{2} + x - 42$.
Find two numbers that multiply to $-42$ and add to $1$. These are $-6$ and $7$.
Express the factored form as $(x - 6)(x + 7)$.
Step 4:
Factor the quadratic $x^{2} - 49$ using the difference of squares.
Recognize $49$ as $7^2$.
Apply the formula $a^2 - b^2 = (a + b)(a - b)$ with $a = x$ and $b = 7$ to get $(x + 7)(x - 7)$.
Step 5:
Factor the quadratic $x^{2} - 11x + 28$.
Find two numbers that multiply to $28$ and add to $-11$. These are $-7$ and $-4$.
Express the factored form as $(x - 7)(x - 4)$.
Step 6:
Simplify by canceling out common factors.
- Cancel out $(x - 4)$ and $(x + 7)$ from the numerator and denominator.
Step 7:
Continue simplifying by canceling out $(x - 6)$ and $(x - 7)$.
Step 8:
Multiply the remaining expression by $1$ to maintain equality.
Step 9:
Factor the numerator $x^{2} - 49$ using the difference of squares.
- Recognize $49$ as $7^2$ and factor as $(x + 7)(x - 7)$.
Step 10:
Simplify the denominator $3x^{2} - 147$.
Factor out $3$ to get $3(x^{2} - 49)$.
Recognize $49$ as $7^2$ and factor as $3(x + 7)(x - 7)$.
Step 11:
Reduce the expression by canceling out $(x + 7)$ and $(x - 7)$.
Step 12:
The final simplified form is $\frac{1}{3}$.
Knowledge Notes:
To solve this problem, several algebraic concepts and techniques are used:
Multiplying by the Reciprocal: When dividing by a fraction, you multiply by its reciprocal. This is based on the property that dividing by a number is the same as multiplying by its inverse.
Factoring Quadratics: Factoring quadratics involves finding two numbers that multiply to give the constant term (c) and add to give the coefficient of the linear term (b). This is often done using the AC method.
Difference of Squares: This is a special factoring technique used when a binomial is in the form of $a^2 - b^2$. It factors into $(a + b)(a - b)$.
Canceling Common Factors: When the same factor appears in both the numerator and the denominator of a fraction, it can be canceled out.
Simplifying Expressions: The process involves reducing fractions to their simplest form by canceling common factors.
In this problem, the steps are carefully structured to gradually simplify the complex algebraic fraction by factoring and canceling, ultimately arriving at the simplest form.
