Simplify (((12y^2-7y-12)/(12y^2+25y+12))/(12y^2-25y+12))/(16y^2-24y+9)

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 $a{x}^2+bx+c$, 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:

1. 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.

2. Reciprocal Multiplication: When dividing by a fraction, you can multiply by its reciprocal (i.e., flip the numerator and denominator).

3. 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.

4. Common Factor Elimination: When a factor appears in both the numerator and the denominator, it can be canceled out to simplify the expression.

5. Power Rule for Exponents: When multiplying like bases, you add the exponents ($a^m\times a^n=a^{m+n}$).

6. Latex Formatting: Mathematical expressions are formatted using Latex to clearly present equations and solutions.