Factor: 7r2 + 8r + 1

Factor the first term:

7r2 = (7r) (r)

7r r

Write them side by side with space

The last term is positive and the middle term is positive

The last term should have two positive factors

+1 = (+1) (+ 1)

Write these below 7r and r

+1 + 1

4.3 More on Factoring Trinomials

7r r

+1 + 1

+1 (7r) = 7 r

+1(r) = r

Adding gives: 7r + r = 8r

Pick the factors vertically

The first factor is (7r + 1)

The second factor is (r + 1)

7r2 + 8r + 1 = (7r + 1) (r + 1)

Multiply diagonally: 7r2 + 8r + 1

Factor: 20x2 + 11x – 3

5x 4x 20x2 can be factored as:

20x2 = (5x) (4x)

-3 = (+3) (-1)

Write +3 below 4x and –1 below 5x

-1 +3

Write this side by side

You need to do this by trial and error

20x2 + 11x – 3

5x 4x

-1 +3

(+3) (5x) = 15x

-1 (4x) = -4x

Adding gives: 15x – 4x = 11x

Pick the factors vertically

The first factor is (5x – 1)

The second factor is (4x + 3)

Multiply diagonally:

20x2 + 11x – 3 = (5x – 1) (4x + 3)

Factor: 48b2 –74b – 10

2 is a common factor

8b 3b

48b2 –74b – 10 = 2 (24b2 –37b – 5 )

24b2 = (8b) (3b)

+ 1 – 5

We now factor inside the parenthesis.

– 5 = (– 5)(+1)

Write –5 below 3b and +1 below 8b

2 (24b2 –37b – 5)

8b 3b

+ 1 – 5(+1) (3b) = 3b

– 5 (8b) = – 40b

Adding gives: – 40b + 3b = -37b

The first factor is (8b + 1)

The second factor is (3b – 5)

Multiply diagonally:

48b2 –74b – 10 = 2(8b + 1)(3b – 5)

Factor: 24a4 + 10a3 – 4a2

2a2 is a common factor

4a 3a

24a4 + 10a3 – 4a2 = 2a2 (12a2 + 5a – 2 )

12a2 = (4a) (3a)

-1 +2

We now factor inside the parenthesis.

– 2 = (+2)(-1)

Write +2 below 3a and -1 below 4a

2a2 (12a2 + 5a – 2 )

4a 3a

-1 +2

(-1) (3a) = -3a

+2 (4a) = + 8a

Adding gives: 8a – 3a = 5a

The first factor is (4a – 1)The second factor is (3a + 2)

Multiply diagonally:

24a4 + 10a3 – 4a2 = 2a2(4a – 1)(3a + 2)

Factor: 18 + 65x + 7x2

9 2 18 = (9) (2)

7x2 = (+7x) (+x)

Write +7x below 2 and +x below 9

+x +7x

Write this side by side

You need to do this by trial and error

18 + 65x + 7x2

9 2

+x +7x

(+7x) (9) = 63x

(x) (2) = 2x

Adding gives: 63x + 2x = 65x

Pick the factors vertically

The first factor is (9 + x)

The second factor is (2 + 7x)

Multiply diagonally:

18 + 65x + 7x2 = (9 +x) (2 + 7x)

=(x + 9)(7x + 2)

Factor: -18k3 – 48k2 + 66k

-6k is a common factor 3k k -18k3 – 48k2 + 66k = -6k (3k2 + 8k – 11 )

3k2 = (3k) (k)

+11 -1We now factor inside the parenthesis.

– 11 = (+11)(-1)

Write -1 below k and +11 below 3k

3k k

+11 -1

(11) (k) = 11k

-1 (3k) = -3k

Adding gives: -3k + 11k = 8k

The first factor is (3k + 11)

The second factor is (k –1)

Multiply diagonally:

-18k3 – 48k2 + 66k = -6k(3k + 11)(k – 1)

-6k (3k2 + 8k – 11 )

Factor: 12k3q4 – 4k2q5 – kq6

kq4 is the common factor

6k 2k

12k3q4 – 4k2q5 – kq6 = kq4 (12k2 – 4kq – q2)

12k2 = (6k) (2k)

+q -q

We now factor inside the parenthesis.

– q2 = (+q)(-q)

Write -q below 2k and +q below 6k

6k 2k

+q -q(q) (2k) = 2kq

-q (6k) = -6kq

Adding gives: -6kq + 2kq = -4kq

The first factor is (6k + q)

The second factor is (2k –q)

Multiply diagonally:

12k3q4 – 4k2q5 – kq6 = kq4(6k + q)(2k – q)

kq4 (12k2 – 4kq – q2)

Factor: 14a2b3 +15ab3 – 9b3

b3 is the common factor

7a 2a

14a2b3 +15ab3 – 9b3 = b3 (14a2 + 15a – 9 )

14a2 = (7a) (2a)

-3 +3

We now factor inside the parenthesis.

– 9 = (+3)(-3)

Write +3 below 2a and -3 below 7a

b3 (14a2 + 15a – 9 )

7a 2a

-3 +3(-3) (2a) = - 6a

3 (7a) = 21a

Adding gives: 21a – 6a = 15a

The first factor is (7a – 3)

The second factor is (2a + 3)

Multiply diagonally:

14a2b3 +15ab3 – 9b3 = b3(7a – 3)(2a + 3)