MTH 030 -- Elementary Algebra -- Exercise Solutions
Section:
5.3

27)  Factor 2x2 - 3x + 1
       Solution:

2x2 - 3x + 1 = (2x - 1)(x - 1)

justify by multiplying
The constant term is positive so the two factors will have the same sign. Because the middle term is negative, the sign in both factors will be -.

28)  Factor 2y2 - 7y + 3
       Solution:  2y2 - 7y + 3 = (2y - 1)(y - 3)

justify by multiplying

The constant term is positive so the two factors will have the same sign. Because the middle term is negative, the sign in both factors will be -.

30)  Factor 2b2 + 7b + 6
       Solution:   2b2 + 7b + 6 = (2b + 3)(b + 2)


justify by multiplying

The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.
32)  Factor 4t2 - 4t + 1
       Solution:  4t2 - 4t + 1 = (2t - 1)(2t - 1)
       = (2t - 1)2


justify by multiplying
 Both the leading term and the constant term are perfect squares and twice the cross product is the middle term. Therefore this trinomial factors as the square of a difference.
34)  Factor 4x2 + 8x + 3
       Solution:  4x2 + 8x + 3 = (2x + 3)(2x + 1)


justify by multiplying
The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.

36)  Factor 4z2 - 9z + 2
       Solution:  4z2 - 9z + 2 = (4z - 1)(z - 2)


The constant term is positive so the two factors will have the same sign. Because the middle term is negative, the sign in both factors will be -.

38)  Factor 8u2 - 2u - 15
       Solution:  8u2 - 2u - 15 = (2u - 3)(4u + 5)

justify by multiplying

The constant term is negative so one factor will have a minus sign the other will have a plus sign.

40)  Factor 12y2 - y - 1
       Solution:   12y2 - y - 1 = (4y + 1)(3y - 1)

justify by multiplying

The constant term is negative so one factor will have a minus sign the other will have a plus sign.

42)  Factor 10u2 - 13u - 6
       Solution:  10u2 - 13u - 6 is prime

justify by multiplying

 Click Here to see all the possible factors

44)  Factor 6m2 + 19m + 3
       Solution:  6m2 + 19m + 3
       = (6m + 1)(m + 3)

justify by multiplying

The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.
46)  Factor 10x2 + 21x - 10
      Solution:  10x2 + 21x - 10
      = (2x + 5)(5x - 2)


justify by multiplying
The constant term is negative so one factor will have a minus sign the other will have a plus sign.
48) Factor - 16y2 - 10y - 1
      Solution:  - 16y2 - 10y - 1 =(-1)(16y2 + 10y + 1)
      = (-1)(8y + 1)(2y + 1) = - (8y + 1)(2y + 1)

justify by multiplying

As a first step factor -1 from the expression.
Then observe the following about the trinomial.

The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.

50) Factor - 16x2 - 16x - 3
      Solution:   - 16x2 - 16x - 3 = (-1)(16x2 + 16x + 3)
      = (-1)(4x + 1)(4x + 3) = -(4x + 1)(4x + 3)

justify by multiplying
As a first step factor -1 from the expression.
Then observe the following about the trinomial.

The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.
52) Factor 2b2 - 5bc + 2c2
      Solution:  2b2 - 5bc + 2c2 = (2b - c)(b - 2c)

justify by multiplying
The constant term is positive so the two factors will have the same sign. Because the middle term is negative, the sign in both factors will be -.
52 Altrernate) Factor 2c2 - 5bc + 2b2
      Solution:  2c2 - 5bc + 2b2 = (2c - b)(c - 2b)

justify by multiplying
The constant term is positive so the two factors will have the same sign. Because the middle term is negative, the sign in both factors will be -.
54) Factor 3m2 + 5mn + 2n2
      Solution:  3m2 + 5mn + 2n2 = (3m + 2n)(m + n)

justify by multiplying
The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.
54 Alternate) Factor 2n2 + 5mn + 3m2
      Solution:  2n2 + 5mn + 3m2 = (2n + 3m)(n + m)

justify by multiplying
The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.
56) Factor 4b2 + 15bc - 4c2
      Solution:  4b2 + 15bc - 4c2 = (4b - c)(b + 4c)

justify by multiplying
The constant term is negative so one factor will have a minus sign the other will have a plus sign.
58) Factor 12x2 + 5xy - 3y2
      Solution:  12x2 + 5xy - 3y2 = (4x + 3y)(3x - y)

justify by multiplying
The constant term is negative so one factor will have a minus sign the other will have a plus sign
60) Factor - 14 + 3a2 - a
      Solution:   - 14 + 3a2 - a = 3a2 - a - 14
      = (3a - 7)(a + 2)

justify by multiplying
As a first step write the trinomial in descending powers of the variable.
Then observe the following about the trinomial.

The constant term is negative so one factor will have a minus sign the other will have a plus sign
62) Factor 16 - 40a + 25a2
      Solution:  16 - 40a + 25a2 = 25a2- 40a + 16
      = (5a - 4)(5a - 4) = (5a - 4)2

justify by multiplying
As a first step write the trinomial in descending powers of the variable.
Then observe the following about the trinomial.

The first and last terms are perfect squares and the middle term is twice the cross product . Therefore the trinomial is a perfect square. Since the middle term is negative, the trinomial is the square of a difference.
64) Factor 12t2 - 1 - 4t
      Solution:  12t2 - 1 - 4t = 12t2 - 4t - 1
      = (6t + 1)(2t - 1)

justify by multiplying
As a first step write the trinomial in descending powers of the variable.
Then observe the following about the trinomial.
The constant term is negative so one factor will have a minus sign the other will have a plus s

66) Factor 25 + 2u2 + 3u
      Solution:  25 + 2u2 + 3u = 2u2 + 3u + 25
      This trinomial is prime.
       Click here to see all the possible factors.

justify by multiplying

As a first step write the trinomial in descending powers of the variable.
Then observe the following about the trinomial.

The constant term is positive so the two factors will have the same sign. Because the middle term is positive, the sign in both factors will be +.

68) Factor 11 uv + 3u2 + 6v2
      Solution:  11 uv + 3u2 + 6v2 = 3u2 + 11 uv + 6v2
       = (3u + 2v)
(u + 3v)

justify by multiplying

70) Factor 12m2 - 1mn + 2n2
      Solution:  12m2 - 1mn + 2n2 = (3m - 2n)(4m - n)


justify by multiplying
72) Factor 9x2 + 21x - 18
      Solution:  9x2 + 21x - 18 = 3(3x2 + 7x - 6)
      = 3(3x - 2)(x + 3)


justify by multiplying
74) Factor - 2xy2 - 8xy + 24x
      Solution:  - 2xy2 - 8xy + 24x =
      = - 2x(y2 + 4y - 12)
      = - 2x(y + 6)(y - 2)

justify by multiplying