There is a pattern for factoring trinomials of this form, when c Factoring Patterns There is a pattern for factoring trinomials of this form, when c is positive x² + bx + c

Trinomials have 3 terms x² + 9x +8 The first term is of degree two and so the term is called a quadratic term The second term is called the linear term The last term is called the constant (it has no variable factor) The trinomial itself is called a quadratic polynomial

Examples of trinomials in this form x² + 9x + 8 r² + 10 r + 24 y² - 14y + 13 m² - 10m + 16 NOTE: coefficient of quadratic term is 1 constant term is positive Notice that in each case the coefficient of the quadratic term is 1 and the constant term is positive.

To factor trinomials like x² + 8x + 16 List pairs of factors whose product equal the constant term 8 2 16 1 4 4 Find the pair of factors whose sum equals the coefficient of the linear term. 4 + 4 = 8

Factor x² + 8x + 16 (x + 4)(x + 4) or (x+4)²

y² + 20y + 36 Factors of 36 1 36 6 6 9 4 12 3 18 2 Sum of factors 37 1 36 6 6 9 4 12 3 18 2 Sum of factors 37 12 13 15 20

So factor y² + 20y + 36 (y + 18) ( y + 2) Check by multiplying the binomials using FOIL y² + 2y + 18y + 36 y² + 20y + 36

Factor x² - 12x + 20 Since the linear term is negative and the constant term is positive we must list the negative factors of 20. -5 -4 -9 -20 -1 -21 -2 -10 -12

So factor x² - 12x + 20 We know the only possible factors are –2 and –10 so we write (x – 2)(x – 10) Check by applying FOIL x² -10x –2x + 20 x² - 12x + 20

Another Factoring Pattern x² - ax – c There is also a pattern for factoring trinomials of this form when c is negative

Trinomials with three terms x² + 29x – 30 m² + 12m – 36 k² - 25 k – 54 g² -g – 2 Note: coefficient of quadratic term is 1 constant term is negative

To factor trinomials like x² + 7x - 18 List pairs of factors of -18 6 -3 -6 3 - 9 2 9 -2 18 -1 -18 1 Sum of factors 3 -3 -7 7 17 -17

So factor x² + 7x - 18 Since the linear term is positive select factors which give a positive result when added. But remember, because the constant term is negative, one factor must be negative. Using the preceding factor list we can write (x + 9) (x – 2) Check using FOIL x² - 2x + 9x – 18 x² + 7x - 18