Algebra I : Semester II : Polynomials

Sections:

Introduction  |   Section 1  |   Section 2  |   Section 3  |  Section 4  |  Section 5 | Section 6

Section Five

Part 1  |  Part 2  |   Part 3

Algebra 1: Section 5: Polynomials Factoring Simple Trinomials   Click here for a presentation on Factoring Simple Trinomials Now, let’s write the rules just learned in the presentation.   How to Factor a Trinomial of the Form x2 + bx + c Write a list of the factors that give a product of c, the last term. In the second column of the table, find the sum of the numbers. When the sum is the same as b, the coefficient of the middle term, write the two as the second term in each binomial, (x + __)(x + __). You may need to change one or both plus signs to minus. Check by using FOIL. The following examples are the same ones used in the presentation. They are repeated here for easy access. Example 1: Factoring x2 + bx + c Factor x2 + 6x + 8. Write a list of the factors that will give a product of eight. In the second column, write their sum. Product of 8 Sum 1, 8 9 2, 4 6 The sum of two and four is the same as the middle term’s coefficient, six. (x + 2)(x + 4) Check:   (x + 2)(x + 4) = x2 + 4x + 2x + 8 = x2 + 6x + 8 Example 2: Factoring x2 – bx + c Factor x2 – 22x + 21. Write a list of the factors that will give a product of 21. In the second column, write their sum. Product of 21 Sum 1, 21 22 3,7 10 Since none of these give a sum of -22, both numbers must be negative. This would still give a product of positive 21.   -1, -21 -22 The sum of -1 and -21 is the same as the middle term’s coefficient, -22. (x – 1)(x – 21) Check:  (x – 1)(x – 21)= x2 – 21x – 1x + 21 = x2 – 22x + 21 Example 3: Factoring x2 – bx – c Factor x2 – x – 20. Write a list of the factors that will give a product of -20. In the second column, write their sum. Product of -20 Sum -1, 20 19 1, -20 -19 -2, 10 8 2, -10 -8 -4, 5 1 4, -5 -1 The sum of four and negative 5 is the same as the middle term’s coefficient, negative one. (x + 4)(x – 5) Check:  (x + 4)(x – 5)= x2 – 5x + 4x – 20  = x2 – x – 20. Example 4: Factoring x2 + bx – c Factor x2 + 3x – 40. Write a list of the factors that will give a product of -40. In the second column, write their sum. Product of -40 Sum -1, 40 39 1, -40 -39 -2, 20 18 2, -20 -18 -4, 10 6 4, -10 -6 -5, 8 3 5, -8 -3 The sum of negative five and eight is the same as the middle term’s coefficient, three. (x – 5)(x + 8) Check:  (x – 5)(x + 8)= x2 +8x – 5x – 40 = x2 + 3x – 40. If you are getting pretty good at finding these mentally, use the short rule below. But remember, when in doubt, write it out.   Short Rule for Factoring x2 + bx + c Look for two numbers that give the product of c and the sum of b. Write them in the binomials, (x + __) (x + __), changing signs if necessary. Here is a rule of thumb if the signs in the binomials are causing you problems.  Signs to Use when Factoring x2 + bx + c = (x + __) (x + __) x2 – bx + c = (x – __) (x – __) x2 + bx – c = (x + __) (x – __) x2 – bx - c = (x + __) (x – __)        OR       Quick Rule When c is positive, the factors will have both + or both –. When c is negative, the factors will have one + and one –.   Practice Click to get a new problem.  Factor the polynomial and then click to see the answer.  Remember that your answer is still correct if you have the same binomials reversed. Now go on to the next part

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