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Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. For example, the polynomial function in the exercise \( x^{2}-5x \) consists of terms \( x^{2} \) and \( -5x \) where \( x \) is the variable, and the corresponding coefficients are 1 and -5, respectively. They are the building blocks of many algebraic equations and can model various phenomena in sciences.

Polygons get their name from the term 'polynomial', where 'poly' means 'many' and 'nomial' means 'terms'. Indeed, polynomials may include many terms, such as \( x^{2} \) (a quadratic term), \( -5x \) (a linear term), or even a constant like 3. These components come together to form a full polynomial function.

In algebra, like terms are terms that contain the same variables raised to the same power. The coefficients can be different. For instance, \( x^{2} \) and \( 6x^{2} \) are like terms because they both contain \( x \) raised to the second power, though they have different coefficients, 1 and 6, respectively.

When subtracting polynomials, like terms are crucial because they dictate which terms can be combined. In our exercise, subtracting \( x^{2} - 6x^{2} \) focuses on combining these like terms since they refer to the same power of \( x \) and simplifying those results in \( -5x^{2} \) after considering the subtraction operation.

Simplifying expressions means to combine like terms and any constants to make an algebraic expression as compact as possible. This process often involves basic arithmetic operations: addition, subtraction, multiplication, and, occasionally, division. In the context of our exercise, we simplify the polynomial subtraction by first distributing the minus sign across the polynomial we are subtracting, effectively changing the signs of its terms.

Then, we combine the like terms by subtracting their coefficients. For the \( x^{2} \) terms, we have 1 - 6, resulting in -5. For the \( x \) terms, we have -5 - (-4), giving us 1. So our simplified expression is \( -5x^{2} + x \) where each term is now as simple as it can be with respect to combining terms. This step is vital for solving equations, simplifying complex expressions, and gaining better understanding of algebraic structures.