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A quadratic expression is a type of polynomial that is characterized by having a variable raised to the second power. In general, a quadratic expression can be written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The term \(ax^2\) is what gives the expression its quadratic nature due to the squared variable. In the expression you are working with, \(c^2 + 19c + 34\), you can see it fits the general form as \(c^2\) is the squared term. Understanding the structure of the quadratic expression is an essential step in learning how to factor it.

Factoring involves breaking down the quadratic into simpler expressions (or factors) that, when multiplied together, result in the original quadratic expression. This method often relies on finding two numbers that accomplish certain relationships with the coefficients of the terms.

Every term in a polynomial expression has a coefficient. These coefficients are crucial because they help determine how the polynomial behaves and how it can be factored. Consider the quadratic expression given: \(c^2 + 19c + 34\). In this context:

• The coefficient of \(c^2\) is 1.
• The coefficient of \(c\) is 19.
• The constant term, represented as 34, is also considered a coefficient in this analysis.

The process of factoring begins with identifying these coefficients. The goal is to find two numbers that multiply to the product of the first and last coefficients while adding to the middle coefficient. The numbers in this case are 2 and 17, both of which satisfy these conditions while corresponding directly to the coefficients of the polynomial.

Once you have identified the two numbers that multiply to the constant term and add up to the middle coefficient, you can transform the quadratic expression into its factored form. For the expression \(c^2 + 19c + 34\), we identified the numbers 2 and 17. They multiply to 34 and together add to 19.

Using these numbers, we can now write the quadratic expression in its factored form as \((c+2)(c+17)\).

• This form is significant because it makes solving quadratic equations easier, particularly when you need to find the roots of the equation where it equals zero.
• Factoring simplifies the equation, providing deeper insights into its solutions and behavior graphically.

Understanding how to find and interpret the factored form is a foundational skill in algebra that facilitates various mathematical processes.