Fundamentals of Mathematics
§4.3 - Factoring Polynomials (Part 2)
Ricky Ng
November 15, 2013
Ricky Ng
Fundamentals of Mathematics
Announcements
1
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2
Test 4 registration begins on Wednesday.
3
Online quiz 13 (last) is due this Friday.
4
Teaching evaluation is available now.
Ricky Ng
Fundamentals of Mathematics
Last Time
Given a trinomial of the form ax2 + bx + c...
1
Test for factorability: If the discriminant D = b2
a perfect square, then it is factorable.
Ricky Ng
Fundamentals of Mathematics
4ac is
Last Time
Given a trinomial of the form ax2 + bx + c...
1
2
Test for factorability: If the discriminant D = b2
a perfect square, then it is factorable.
To factor when a = 1:
4ac is
(i) List the factors of c in pairs, say u and v.
(ii) Change the signs of u and v, according to the sign of c.
(iii) Check which combination gives u + v = b.
Ricky Ng
Fundamentals of Mathematics
Last Time
Given a trinomial of the form ax2 + bx + c...
1
2
Test for factorability: If the discriminant D = b2
a perfect square, then it is factorable.
To factor when a = 1:
4ac is
(i) List the factors of c in pairs, say u and v.
(ii) Change the signs of u and v, according to the sign of c.
(iii) Check which combination gives u + v = b.
3
When a = 1, a direct shortcut to find u and v is
p
p
D b
D b
u=
and
v=
.
2
2
This helps when c has many factors.
Ricky Ng
Fundamentals of Mathematics
Popper 25: Question 1
Determine if
x2
5x
6
is factorable. If so, factor it.
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Popper 25: Question 1
Determine if
x2
5x
6
is factorable. If so, factor it.
a)
b)
c)
d)
e)
Not factorable.
Factorable with
Factorable with
Factorable with
Factorable with
D
D
D
D
= 49; (x 3)(x 2).
= 1; (x 6)(x + 1).
= 1; (x 3)(x + 2).
= 49; (x 6)(x + 1).
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
How to factor when a 6= 1?
Now when a 6= 1, there is no more easy shortcut.
Rule
To factor ax2 + bx + c, we follow a similar scheme:
(i) List the factors of ac in pairs, say u and v.
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
How to factor when a 6= 1?
Now when a 6= 1, there is no more easy shortcut.
Rule
To factor ax2 + bx + c, we follow a similar scheme:
(i) List the factors of ac in pairs, say u and v.
(ii) Change signs of u and v, according to the sign of c.
(iii) Check which combination gives u + v = b.
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
How to factor when a 6= 1?
Now when a 6= 1, there is no more easy shortcut.
Rule
To factor ax2 + bx + c, we follow a similar scheme:
(i) List the factors of ac in pairs, say u and v.
(ii) Change signs of u and v, according to the sign of c.
(iii) Check which combination gives u + v = b.
(iv) Rewrite
ax2 + bx + c = ax2 + ux + vx + c,
then factor by grouping.
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Examples
Determine if the following trinomials are factorable. If so, factor
it.
2x2 + 11x + 9
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
6x2
x
2
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Popper 25: Question 2
The trinomial
5x2 + 13x
6
is factorable. Factor it.
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Popper 25: Question 2
The trinomial
5x2 + 13x
6
is factorable. Factor it.
a)
b)
c)
d)
(5x 1)(x + 6)
(5x + 3)(x 2)
(5x 2)(x + 3)
(5x + 1)(x 6)
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Popper 25: Question 3
The trinomial
3x2 + x
4
is factorable. Factor it.
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
Popper 25: Question 3
The trinomial
3x2 + x
4
is factorable. Factor it.
a)
b)
c)
d)
(3x 2)(x + 2)
(3x + 1)(x 4)
(3x + 2)(x 2)
(3x + 4)(x 1)
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)
More Factoring
Remark
So far we have only learning how to factor ax2 + bx + c and
some with special patterns. Sometimes, it is helpful to first
factor a polynomial by GCF, then apply techniques we have
learned.
Ricky Ng
§4.3 - Factoring Polynomials (Part 2)