Factor 108y-180xy+75x^2y
The question asks for the process called factoring, which involves breaking down a mathematical expression into simpler components or 'factors'. Specifically, it requests to find the factors of the polynomial 108y - 180xy + 75x^2y. In other words, the problem is asking to rewrite the given polynomial as a product of two or more simpler polynomials. The solution would involve finding common multiples and applying algebraic techniques to simplify the expression.
$108 y - 180 x y + 75 x^{2} y$
Answer
Solution:
Step 1: Extract the common factor
Extract the common factor $3y$ from the expression $108y - 180xy + 75x^2y$.
Step 1.1: Factor out $3y$ from the first term
Take $3y$ out of $108y$: $3y(36) - 180xy + 75x^2y$.
Step 1.2: Factor out $3y$ from the second term
Take $3y$ out of $-180xy$: $3y(36) - 3y(60x) + 75x^2y$.
Step 1.3: Factor out $3y$ from the third term
Take $3y$ out of $75x^2y$: $3y(36) - 3y(60x) + 3y(25x^2)$.
Step 1.4: Combine the terms with the common factor $3y$
Combine the terms to factor $3y$ out completely: $3y(36 - 60x + 25x^2)$.
Step 2: Factor the quadratic expression
Identify and factor the quadratic expression inside the parentheses.
Step 2.1: Express $36$ as a square
Write $36$ as $6^2$: $3y(6^2 - 60x + 25x^2)$.
Step 2.2: Express $25x^2$ as a square
Write $25x^2$ as $(5x)^2$: $3y(6^2 - 60x + (5x)^2)$.
Step 2.3: Verify the middle term
Ensure that the middle term $-60x$ is twice the product of the square roots of the first and third terms: $60x = 2 \cdot 6 \cdot (5x)$.
Step 2.4: Rewrite the quadratic expression
Rewrite the expression as a perfect square: $3y(6^2 - 2 \cdot 6 \cdot (5x) + (5x)^2)$.
Step 2.5: Apply the perfect square trinomial formula
Factor the trinomial using the formula $a^2 - 2ab + b^2 = (a - b)^2$, where $a = 6$ and $b = 5x$: $3y((6 - 5x)^2)$.
Knowledge Notes:
To factor a polynomial, one of the strategies is to first look for a common factor in all terms. In this case, the common factor is $3y$, which is present in each term of the polynomial $108y - 180xy + 75x^2y$.
The process of factoring involves rewriting each term of the polynomial such that the common factor is outside of the parentheses, and the remaining expression is inside the parentheses. This is achieved by dividing each term by the common factor.
After factoring out the common factor, the remaining expression inside the parentheses is a quadratic trinomial, which can sometimes be factored further if it represents a perfect square. A perfect square trinomial is an expression of the form $a^2 - 2ab + b^2$, which can be factored into $(a - b)^2$.
To identify a perfect square trinomial, you can check if the first and third terms are perfect squares and if the middle term is twice the product of the square roots of the first and third terms. If these conditions are met, the trinomial can be factored into a binomial squared.
In the given problem, after factoring out $3y$, the remaining quadratic expression is a perfect square trinomial, which can be factored using the perfect square trinomial formula.
