Factoring: b & c Positive

When factoring trinomials, you might encounter

x² + bx + c

when b is positive and c is negative or

x2 + bx + c

when b is negative and c is positive. Here, we''ll look at the process of factoring a trinomial when both b and c are positive.

How a trinomial is formed

First, we''ll examine how a trinomial (a polynomial with exactly 3 terms) is created with the following expression:

(y + 3)(y + 7)

Use FOIL to complete the process:

(y + 3)(y + 7) =

y * y + y * 7 + 3 * y + 3 * 7

= y² + 7y + 3y + 21

= y² + 10y + 21

There is your trinomial. Now, let''s look at how to reverse the process by factoring it.

How to factor a trinomial when b and c are positive

When factoring trinomials in which both b and c are positive, you''ll want to locate two numbers that, when multiplied together, yield the last term (the constant) of the trinomial and, when added together, yield the coefficient of the middle term.

Let''s factor the following trinomial:

x² + 7x + 12

First, we''ll find two numbers whose sum is 7 and product is 12. Let''s list the pairs:

(1, 12) -> 1 x 12 = 12, 1 + 12 = 13(2, 6) -> 2 x 6 = 12, 2 + 6 = 8(3, 4) -> 3 x 4 = 12, 3 + 4 = 7

We''ve located the pair!

Now, let''s rewrite the expression:

x² + 7x + 12

= x² + 3x + 4x + 12

by re-writing

3x + 4x

= (x² + 3x) + (4x + 12)

split the expression into two groups

= x(x + 3) + 4(x + 3)

factor each part using the Distributive Property

= (x + 4)(x + 3)

use the Distributive Property again to extract the factor

(x + 3)

Therefore, the trinomial:

x² + 7x + 12

in factored form is:

(x + 4)(x + 3).

It''s good to note that some trinomials cannot be factored. In other words, they are irreducible polynomials. For instance:

x²+ 17x + 81

cannot be factored because there are no two numbers that can be added to 17 and multiplied to 81.

Practice problems on factoring trinomials when b and c are positive

a. Factor

n² + 13n + 36

(4,9) -> 4 x 9 = 36, 4 + 9 = 13

n² + 4n + 9n + 36 = (n² + 4n) + (9n + 36) = n(n + 4) + 9(n + 4) = (n + 9)(n + 4)

b. Factor:

y² + 17y + 52

Find our pair:

(4,13) -> 4 x 13 = 52, 4 + 13 = 17

y² + 4y + 13y + 52 = (y² + 4y) + (13y + 52) = y(y + 4) + 13(y + 4) = (y + 13)(y + 4)

c. Factor:

x² + 19x + 88

Find our pair:

(8,11) -> 8 x 11 = 88, 8 + 11 = 19

x² + 8x + 11x + 88 = (x² + 8x) + (11x + 88) = x(x + 8) + 11(x + 8) = (x + 11)(x + 8)

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Delve deeper into factoring trinomials when b and c are positive

Students learning how to factor trinomials have a lot of challenging steps to remember. They might struggle with locating two numbers that multiply to create the constant and add to the coefficient -- or they could have trouble using the Distributive Property. Whether your student needs help grasping how to factor trinomials, or they want to boost their knowledge while studying for a test or working on assignments, tutoring can make a significant difference. Learn all about the perks of studying alongside a qualified private educator by reaching out to the Educational Directors at Varsity Tutors today.