Lesson 4: Factoring Trinomials of the Form ax^2 + bx + c: Summary

Math 10C Module 3 Lesson 4

Module 3: Polynomials

Lesson4 Summary

In Lesson 4 you investigated the following questions:

How is a multiplication array similar to the basis of many codes?
How are the coefficients of a trinomial related to the coefficients of its binomial factors?

In this lesson you learned how to factor trinomials of the form ax2 + bx + c, where a ≠ 1. Just as you can use a multiplication array to multiply binomials (see Lesson1), you can also use the arrays to factor trinomials.

In this way, the multiplication array is similar to a cipher. To encode a message, for example, the letters of the plaintext are scrambled in a prescribed way. To decode the message, the letters of the ciphertext must be de-scrambled in a way that is precisely the reverse of how it was encoded.

Likewise, you discovered in this lesson that in order to factor a trinomial, you must first break the x-term into two terms. Those terms are then placed centrally in the array. Then you determine the greatest common factors of each column and row to obtain the factors along the top row and left column. This procedure is precisely the reverse of the binomial multiplication procedure using the same array.

Besides using arrays, decomposition can be used to factor trinomials. With decomposition, you can “decompose” or break apart the middle term based on the determination of two numbers. For the trinomial ax2 + bx + c, these numbers must have a product equal to a × c and a sum equal to b.

Another way to factor a trinomial is by inspection. This strategy is based on educated guesswork. However, knowledge of how the coefficients of the trinomial relate to the coefficients of the binomial helps you use this factoring strategy effectively. For example, the following expression shows the proper factoring of the trinomial 2x2 - 7x - 4:

2x2 – 7x – 4 = (2x + 1)(x – 4)

Note that the coefficient of the x2-term in the trinomial is equal to the product of the first terms (or x-terms) of the binomials. Also notice that the constant –4 is equal to the product of the constants in the binomials: 1 × (–4).

In the next lesson you will consider ways to factor special polynomials. These polynomials are deemed special because there are ways to factor them based on interesting patterns. You will have the opportunity to discover these patterns before exploring how these patterns can be developed into factoring strategies.