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Polynomials are algebraic expressions that consist of variables and coefficients, connected by operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. In simplest terms, they look like sums of terms. For example, in the expression \(y^3 - 10y^2 + 25y\), each part like \(y^3\), \(-10y^2\), and \(25y\) is called a term. These terms can have variables raised to a power, and they are combined based on arithmetic operations. Polynomials can have one or more variables, but in this case, we are focusing on just one variable, \(y\). Key things to note about polynomials:

• They are often ordered by the degree of the variable, meaning the largest exponent term comes first.
• The degree of a polynomial is crucial as it tells how complex the graph or structure of the polynomial might be.

In factoring, understanding the structure of your polynomial is essential because it guides you on which methods to apply.

Factoring starts with identifying the greatest common factor (GCF) in all terms of the polynomial. A common factor is a number or variable shared by all terms in an expression. In the polynomial \(y^3 - 10y^2 + 25y\):

• Each term contains the variable \(y\).
• Thus, \(y\) is the common factor for all terms.

Once the common factor is identified, we can factor it out of the entire expression. Remove \(y\) from each term, which simplifies the polynomial to \(y(y^2 - 10y + 25)\). This step is crucial because it reduces complexity and can reveal further factorizable forms within the expression. Always check if additional factoring steps are possible after removing the common factor.

A perfect square trinomial is a special kind of quadratic expression that results when a binomial is squared. The form looks like \(a^2 + 2ab + b^2\), which can be factored down to \((a + b)^2\). In the expression \(y^2 - 10y + 25\), we need to determine if it's a perfect square trinomial:

• We notice that this trinomial matches the pattern: \(a^2 = y^2\), \(2ab = -10y\), and \(b^2 = 25\).
• Solving gives us \(a = y\) and \(b = 5\), since \((-5) imes 2 = -10\) and \((-5)^2 = 25\).
• This means \(y^2 - 10y + 25\) is indeed a perfect square trinomial, thus it can be factored as \((y - 5)^2\).

Spotting these patterns helps break down complex polynomials into simpler components, aiding in their complete factorization.