Quadratic Trinomials
A quadratic trinomial is a polynomial of the form ax2 +
bx + c,
a ≠ 0 where a re presents the coefficient of the squared (second degree)
term, b represents the coefficient of the linear (first degree) term and
c represents the constant.
When the trinomial is written in standard form (or descending order
of degree), the coefficient of the squared term is called the leading
coefficient.
Trinomials of the Form x2 + bx + c
Step s.html">Factoring a Trinomial of the Form x2 + bx + c
Step 1: Find the pair of integers whose product is c and whose
sum is b . That is, determine m and n such that mn = c
and m + n = b.
Step 2: Write x2 + bx + c = ( x + m)(x + n).
Step 3: Check your work by multiplying the binomials using the
FOIL method. |
Trinomials of the Form x2 + bx + c
Example: Factor: x2 + 8x + 15
x2 + 8x + 15 = (x + ?)(x + ?)
Find two numbers that we 3 ×
5 = 15
can multiply together to get
15 and add together to get 8. 3 + 5 = 8
x2 + 8x + 15 = (x + 3)(x + 5)
Check: (x + 3)(x + 5) = x2 + 8x + 15 
Trinomials of the Form x2 + bx + c, c < 0
Example: Factor: x2 + x – 42
Find two numbers that we can
multiply together to get – 42
and add together to get 1.
( x2 + x – 42 = (x + 7)(x − 6)
Check: (x + 7)(x – 6 ) = x2 + x – 42 
Trinomials of the Form x2 + bx + c
Form Signs of m and n Example
x2 + bx + c, where b and c
| Form |
Signs of m and n |
Example |
x2 + bx + c, where b and c
are both positive |
m and n are both positive |
x2 + 3x + 2 = (x + 2)(x + 1) |
x2 + bx + c, where b is
negative and c is positive |
m and n are both
negative |
a2 – 7a + 12 = (a – 4)(a – 3) |
x2 + bx + c, where b is
positive and c is negative |
m and n are opposite in
sign and the factor with
the larger absolute value
is positive |
y2 + 2y – 24 = (y + 6)(y – 4) |
x2 + bx + c, where b and c
are both negative |
m and n are opposite in
sign and the factor with
the larger absolute value
is negative |
b2 – 4b – 21 = (b – 7)(b + 3) |
Prime Polynomials
A polynomial that cannot be written as the product of two
other
polynomials (other than 1 or – 1) is said to be a prime polynomial.
Example: Factor: 5x2 − x − 2
The Factors of 5 are:
1 and 5 |
The Factors of 2 are:
1 and 2 |
Possible Factors:
(x – 1)(5x + 2)
(x – 2)(5x + 1) |
 |
The polynomial 5x2 − x − 2 is prime.
Trinomials with a Common Factor
Example: Factor: 2x2 – 32x + 96
2x2 – 32x + 96 = 2(x2 – 16x + 48)
The common factor of 2 can be factored out.
x2 – 16x + 48= ( x – ?)(x – ?)
Find two numbers that we can
multiply together to get 48
and add together to get – 16.
2x2 – 32x + 96 = 2(x – 12)(x − 4)
Check:
2(x – 12)(x − 4)
= 2(x2 – 16x + 48)
= 2x2 – 32x + 96
Negative Leading Coefficients
Example: Factor: – x2 – 12x – 36
Find two numbers that we
can multiply together to get
36 and add together to get 12.
– 1(x2 + 12x + 36)
= – 1(x + 6)(x + 6)
= –(x + 6)2
Check:
–(x + 6)2
= – 1(x + 6)(x + 6)
= – x2 – 12x – 36
Practice
Factor each trinomial completely. If it cannot be factored,
say it is prime.
a2 − 2a − 8
x2 − 9x + 18
x2 + 6x + 8
n2 + 4n − 5
x2 − 5xy + 6y2
2a2 − 14a − 16
3z3 + 3z2 − 18z
−5x2 − 45x − 70
16x2 − 20x3 + 4x4
−15x2 −3x3 − 18x
2x3 y + 4x2 y2 − 6xy3
x2 − 4x + 5
