Simplify (((12y^2-7y-12)/(12y^2+25y+12))/(12y^2-25y+12))/(16y^2-24y+9)
This algebraic expression is a complex rational expression that involves multiple polynomial terms. The question is asking for the simplification of a division of two fractions ((numerator)/(denominator)), where both the numerator and the denominator are themselves fractions composed of polynomials in terms of $y$. The overall expression is nested with operations including division, multiplication (implicit in the division of polynomials), and subtraction of polynomial terms. The objective is to simplify this expression to its simplest form by performing the necessary mathematical operations according to the algebraic rules pertaining to polynomials and complex fractions.
$\frac{\frac{\frac{12 y^{2} - 7 y - 12}{12 y^{2} + 25 y + 12}}{12 y^{2} - 25 y + 12}}{16 y^{2} - 24 y + 9}$
Answer
Solution:
Step 1:
Invert the denominator and multiply it with the numerator: $\frac{\left(\frac{12y^2 - 7y - 12}{12y^2 + 25y + 12}\right)}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 2:
Combine the fractions: $\frac{12y^2 - 7y - 12}{12y^2 + 25y + 12} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 3:
Begin factoring by grouping.
Step 3.1:
For the quadratic $ax^2 + bx + c$, split the middle term into two terms with a product of $ac = 12 \times -12 = -144$ and a sum of $b = -7$.
Step 3.1.1:
Extract $-7$ from $-7y$: $\frac{12y^2 - 7(y) - 12}{12y^2 + 25y + 12} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 3.1.2:
Decompose $-7$ into $9 - 16$: $\frac{12y^2 + (9 - 16)y - 12}{12y^2 + 25y + 12} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 3.1.3:
Distribute the terms: $\frac{12y^2 + 9y - 16y - 12}{12y^2 + 25y + 12} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 3.2:
Extract the greatest common factor (GCF) from each group.
Step 3.2.1:
Group the terms: $\frac{(12y^2 + 9y) - (16y + 12)}{12y^2 + 25y + 12} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 3.2.2:
Factor out the GCF from each group: $\frac{3y(4y + 3) - 4(4y + 3)}{12y^2 + 25y + 12} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 3.3:
Factor out the common factor $4y + 3$: $\frac{(4y + 3)(3y - 4)}{12y^2 + 25y + 12} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 4:
Repeat the factoring by grouping process for the remaining polynomials.
Step 5:
Eliminate the common factor $4y + 3$: $\frac{(3y - 4)}{3y + 4} \times \frac{1}{12y^2 - 25y + 12} \times \frac{1}{16y^2 - 24y + 9}$
Step 6:
Factor by grouping for the polynomial $12y^2 - 25y + 12$.
Step 7:
Simplify by canceling out the common factor $3y - 4$: $\frac{1}{3y + 4} \times \frac{1}{4y - 3} \times \frac{1}{16y^2 - 24y + 9}$
Step 8:
Factor the perfect square $16y^2 - 24y + 9$.
Step 9:
Combine the fractions: $\frac{1 \times 1}{(3y + 4)(4y - 3)(4y - 3)^2}$
Step 10:
Simplify the expression by combining like terms.
Step 11:
Finalize the simplification: $\frac{1}{(3y + 4)(4y - 3)^3}$
Knowledge Notes:
Factoring by Grouping: This technique involves rearranging terms in a polynomial and factoring out the greatest common factor from each group to simplify the expression.
Reciprocal Multiplication: When dividing by a fraction, you can multiply by its reciprocal (i.e., flip the numerator and denominator).
Perfect Square Trinomial: A polynomial of the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$, which can be factored into $(a + b)^2$ or $(a - b)^2$ respectively.
Common Factor Elimination: When a factor appears in both the numerator and the denominator, it can be canceled out to simplify the expression.
Power Rule for Exponents: When multiplying like bases, you add the exponents ($a^m \times a^n = a^{m+n}$).
Latex Formatting: Mathematical expressions are formatted using Latex to clearly present equations and solutions.
