When we speak of trinomials,we refer to polynomials that consist three terms. while quadratic means a polynomial with a degrees of 2.
We will learn how to factor each expression, but in order to do so, let us factor the following expression using the concepts and skills we have learned so far in this book.
Factoring Quadratic Trinomials
The polynomial x^2 + rx + sx + rs is generally characterized as a quadratic polynomial since its degree is 2.
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Quick Check: How do we check if the factors are correct for a polynomial? What operation should be performed?
As illustrated above, the factors of x^2 + rx + sx + rs are ( x + r)( x + s). We can still describe properties of this polynomial through the result of the previous activity. Its leading coefficient is 1, and as for this polynomial, it does not have a GCF except for 1. Each of the factors obtained is a sum of two monomials. Furthermore, by substituting values to r and s, we can simplify the polynomial and get three terms. Let us take for example substituting the values r=2 and s=3.
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In this example, if r = 2 and s = 3, the factors would be (x + 2) (x + 3). Observe the middle term and tge third term. How do the values of r and s relate to the numerical coefficients of these terms? As we can see, the numerical coefficient of tge middle term (5) is the sum of 2 and 3; while the third term is (6) is the product of 2 and 3. We can check this for any value we give to r and s. We can conclude now that a quadratic trinomial, with a leading coefficient of 1, in the form of x^2 + (r + s ) x + rs will have factors of the form (x + r)(x + s).
Examples:
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