Factoring General Trinomials
A trinomial is a 3 term polynomial. For
2
example, 5𝑥 − 2𝑥 + 3 is a trinomial.
In many applications in mathematics, we
need to solve an equation involving a
trinomial. Factoring is an important part
of this process.
Example 1
Factor:
𝑥2 − 5𝑥 − 6
Note that the term we have at the beginning of the
question is x2, so we put x in each bracket. This gives
us (x)(x) = x2, which is what we need for that first term.
Now we need 2 numbers that multiply to give −6 and add to
give −5. The possibilities are:
Factor: 𝑥2 − 2𝑥 − 15
Rational Algebraic
Expression
Rational Algebraic Expression
- The rational algebraic expression is
the quotient of two polynomials
-A rational algebraic expression (or
𝑎
rational expression) is in the form ,
𝑏
where 𝑎 and 𝑏 are both polynomials
and 𝑏 ≠ 0.
5
8
1. is a rational algebraic expression whose
numerator is 5 and denominator is 8. Both
numerator and denominator are monomials.
𝑥
5
2. is a rational algebraic expression whose
numerator can be any rational value. Both
numerator and denominator are monomials.
𝑥
𝑥−5
3.
is a rational algebraic expression. The
numerator is a monomial. the denominator is
a binomial where 𝑥 ≠ 5. the value of x should
not be equal to 5 as it will make the expression
undefined.
𝑥−3
𝑥 2 +5𝑥+6
4.
is a rational algebraic expression in
which both numerator and denominator are
polynomials.
Let us find the excluded value of a rational algebraic
expression. An excluded value means the value of the
denominator that will make it equal to zero.
Examples:
3
1.
𝑥
2.
2
𝑥−7
𝑥≠0
equate the denominator with 0
𝑥−7=0
𝑥=7
Therefore, the excluded value of
2
𝑥−7
is 7.
3.
𝑥
𝑥+5
equate the denominator with 0
𝑥+5=0
𝑥 = −5
Therefore, the excluded value of
𝑥
𝑥+5
is −5
4.
5𝑥
𝑥 2 +4𝑥+3
equate the denominator with 0
𝑥 2 + 4𝑥 + 3 = 0
Factor the general trinomial
𝑥+3 𝑥+1 =0
Use the zero product property to get the value of x.
𝑥+3=0
𝑥 = −3
𝑥+1=0
𝑥 = −1
Therefore, the excluded value of
5𝑥
𝑥 2 +4𝑥+3
are −3 and −1
Simplifying Rational Expressions
Simplifying a rational expression means finding its
lowest term.
Remember when you are finding the lowest term of
a fraction, you have to get the greatest common
factor of the numerator and denominator. The same
process is used in simplifying rational expressions
when both numerator and denominator are
monomials
Examples:
16
1.
72
Find the GCF of 16 and 72. Divide it for both
numerator and denominator.
16
72
8
8
÷ =
2
9
2.
15𝑥 8
25𝑥 6
The GCF of 15𝑥 8 and 25𝑥 6 is 5𝑥 6
Divide the GCF for both numerator and
denominator
8
15𝑥
6
5𝑥
25𝑥
6
5𝑥
÷
6
=
2
3𝑥
5
24𝑥 7 𝑦 6
3.
40𝑥 2 𝑦 10
The GCF of the numerator and denominator is
8𝑥 2 𝑦 6
Divide it for both numerator and denominator.
7 6
2 6
5
24𝑥 𝑦
8𝑥 𝑦
3𝑥
÷ 2 6= 4
2
10
40𝑥 𝑦
8𝑥 𝑦
5𝑦
4𝑥+20
4. 2
𝑥 +7𝑥+10
Factor the 4𝑥 + 20 using common monomial factor
Factor the 𝑥 2 + 7𝑥 + 10
=
4(𝑥+5)
(𝑥+5)(𝑥+2)
cancel the common factors
=
4(𝑥+5)
(𝑥+5)(𝑥+2)
=
4
𝑥+2