Transcript 4.2

4.2
Factoring Trinomials of the
2
Form x + bx + c
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Factoring Trinomials of the Form
x2 + bx + c
Recall by using the FOIL method that
F
O
I
L
(x + 2)(x + 4) = x2 + 4x + 2x + 8
= x2 + 6x + 8
To factor x2 + bx + c into (x + one #)(x + another #),
note that b is the sum of the two numbers and c is the
product of the two numbers.
So we’ll be looking for 2 numbers whose product is c
and whose sum is b.
Note: there are fewer choices for the product, so that’s
why we start there first.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
2
Factoring Trinomials of the Form
x2 + bx + c
To Factor a Trinomial of the Form x2 + bx + c
The product of these numbers is c.
x2 + bx + c = (x + )(x + )
The sum of these numbers is b.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
3
Factoring Polynomials
Example
Factor the polynomial x2 + 13x + 30.
Since our two numbers must have a product of 30 and a
sum of 13, the two numbers must both be positive.
Positive factors of 30
Sum of Factors
1, 30
31
2, 15
17
3, 10
13
Note, there are other factors, but once we find a pair
that works, we do not have to continue searching.
So x2 + 13x + 30 = (x + 3)(x + 10).
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
4
Factoring Polynomials
Example
Factor the polynomial x2 – 11x + 24.
Since our two numbers must have a product of 24 and a
sum of -11, the two numbers must both be negative.
Negative factors of 24
Sum of Factors
– 1, – 24
– 25
– 2, – 12
– 14
– 3, – 8
– 11
So x2 – 11x + 24 = (x – 3)(x – 8).
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
5
Factoring Polynomials
Example
Factor the polynomial x2 – 2x – 35.
Since our two numbers must have a product of – 35 and a
sum of – 2, the two numbers will have to have different signs.
Factors of – 35
Sum of Factors
– 1, 35
34
1, – 35
– 34
– 5, 7
2
5, – 7
–2
So x2 – 2x – 35 = (x + 5)(x – 7).
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
6
Prime Polynomials
Example
Factor: x2 – 6x + 10
We look for two numbers whose product is 10 and
whose sum is – 6. The two numbers will have to
both be negative.
Negative factors of 10 Sum of Factors
– 1, – 10
– 11
– 2, – 5
–7
Since there is not a factor pair whose sum is – 6,
x2 – 6x +10 is not factorable and we call it a prime
polynomial.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
7
Check Your Result!
You should always check your factoring
results by multiplying the factored polynomial
to verify that it is equal to the original
polynomial.
Many times you can detect computational
errors or errors in the signs of your numbers
by checking your results.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
8