May 25, 2010

Algebra: Root of a Linear Equation With One Unknown (Non-Fractional



1. 4x + 5 = 7x - 10
Answer: x = 5
2. 3x - 4 = 8x + 41
Answer: x = -9
3. 11x + 7 = 122 - 12x
Answer: x = 5
4. 9x + 7 = 187 - 9x
Answer: x = 10
5. 10x + 7 = 7x - 26
Answer: x = -11
6. x + 3 = 7x + 21
Answer: x = -3
7. 8x + 7 = 37 - 2x
Answer: x = 3
8. 3x + 11 = 9x - 13
Answer: x = 4
9. 2x + 3 = 58 - 3x
Answer: x = 11
10. 11x + 5 = 29 + 7x
Answer: x = 6
11. 4x - 6 = 7 - 13x
Answer: x = 13/17
12. x - 4 = 5 - 5x
Answer: x = 3/2
13. 14x + 5 = 5x + 9
Answer: x = 4/9
14. 11x - 2 = 8x - 4
Answer: x = -2/3
15. 6x + 2 = 7 - 14x
Answer: x = 1/4
16. 4x - 13 = 14 - 11x
Answer: x = 9/5
17. 4x - 3 = 4 - 10x
Answer: x = 1/2
18. 3x + 10 = 15 - 12x
Answer: x = 1/3
19. 5x + 7 = 10 - 10x
Answer: x = 1/5
20. 11x - 1 = 2 - 13x
Answer: x = 1/8
Posted by Gerardo Emerzona I

May 9, 2010

Algebra: Graphing of Linear Equation Problems













Posted by Gerardo Emerzona I

April 22, 2010

Algebra: The Polynomial Arithmetic



The following sets of problems are about:

  • Addition of Polynomials
  • Subtraction of Polynomials
  • Multiplication of Polynomials
  • Division of Polynomials

Addition of Polynomials

1.9x - y - 2z and 7x + 5y - 8z
Answer: 16x + 4y - 10z
2.8m - 7n + 7 and 5m + 3n - 2
Answer: 13m - 4n + 5
3.5a + 3b + 9 and 2a - 9b + 7
Answer: 7a - 6b + 16
4.9a + 4b + 9c and 3a + 3b + 2c
Answer: 12a + 7b + 11c
5.6x - 7y - 6z, 9x + 8y - 7z and 9x - 5y - 7z
Answer: 24x - 4y - 20z
6.2x - 9y + 2z, 6x - 5y + z and 7x + 6y + 9
Answer: 15x - 8y + 3z + 9
7.5a + 6b - 3, 1a + b - 6c and 3a - 7b + 5
Answer: 9a - 6c + 2
8.6x - 7y - 6z, 9x + 8y - 7z and 9x - 5y - 7z
Answer: 24x - 4y - 20z
9.2x - 9y + 2z, 6x - 5y + z and 7x + 6y + 9
Answer: 15x - 8y + 3z + 9
10.3a + 5b + 9, 8a - 9b - 7c and 9a + b - 3
Answer: 20a - 3b - 7c + 6
11.6a2 - 6ab - 12a and 7a2 - 28ab - 7a
Answer: 13a2 - 34ab - 19a
12.14a2b + 56ab2 + 21ab and 42a2b - 7ab2 - 7ab
Answer: 56a2b + 49ab2 + 14ab
13.3a5 + 5a4b + 8a4 and 20a5 + 35a4b - 10a4
Answer: 23a5 + 40a4b - 2a4
14.35x2 + 28xy + 21x and 20x2 + 4xy - 28x
Answer: 55x2 + 32xy - 7x
15.10x2y + 5xy2 + 40xy and 49x2y + 14xy2 + 7xy
Answer: 59x2y + 19xy2 + 47xy
16.49x5 - 35x4y + 49x4 and 8x5 + 14x4y + 14x4
Answer: 57x5 - 21x4y + 63x4
17.4x4 + x3 + 9 and 2x4 + 7x3 + 3
Answer: 6x4 + 8x3 + 12
18.8m3 + m2 - 6 and 7m3 + 8m2 - 7
Answer: 15m3 + 9m2 - 13
19.4b2 + 9b + 5 and 2b2 + 5b + 2
Answer: 6b2 + 14b + 7
20.63x2 - 9xy - 63x and 30x2 - 40xy - 45x
Answer: 93x2 - 49xy - 108x


Subtraction of Polynomials

1.9x + 2y + 3z and 7x + 6y + 8z
Answer: 2x - 4y - 5z
2.2m + 8n + 2 and m - 7n - 8
Answer: m + 15n + 10
3.8a + 2b - 1 and 7a - b - 3
Answer: a + 3b + 2
4.6a + 5b + 8c and 4a - 7b + 7c
Answer: 2a + 12b + c
5.6x - 3y + 8z, x - 3y - 5z and 2x + 7y - 2z
Answer: 3x - 7y + 15z
6.9x + 4y + 9z, 4x - 7y + z and x + 5y - 2
Answer: 4x + 6y + 8z + 2
7.9a + 3b - 5, 8a + 7b + 4c and a - b + 4
Answer: - 3b - 4c - 9
8.8x - 4y + 5z, 3x + 4y - 7z and 5x + 4y - z
Answer: -12y + 13z
9.9x - 5y - 9z, 2x + 2y - 3z and 5x + 6y - 5
Answer: 2x - 13y - 6z + 5
10.9a + 3b - 8, a + 5b + 2c and 7a - 8b + 3
Answer: a + 6b - 2c - 11
11.4a2 - 20ab - 4a and a2 - 8ab + 3a
Answer: 3a2 - 12ab - 7a
12.63a2b + 36ab2 - 45ab and 9a2b - ab2 - 5ab
Answer: 54a2b + 37ab2 - 40ab
13.24a5 - 21a4b - 24a4 and 20a5 + 35a4b - 45a4
Answer: 4a5 - 56a4b + 21a4
14.24x2 + 24xy + 9x and 12x2 + 24xy - 4x
Answer: 12x2 + 13x
15.21x2y + 42xy2 + 7xy and 9x2y + 9xy2 + 63xy
Answer: 12x2y + 33xy2 - 56xy
16.18x3 - 63x2y - 81x2 and 6x3 - 5x2y - 8x2
Answer: 12x3 - 58x2y - 73x2
17.9x2 + 2x + 7 and 4x2 + 3x + 8
Answer: 5x2 - x - 1
18.9m2 - 7m + 6 and 8m2 - 2m + 1
Answer: m2 - 5m + 5
19.8b2 - 3b + 1 and 7b2 - b + 5
Answer: b2 - 2b - 4
20.5x2 - 7x + 8 and 4x2 - 5x + 7
Answer: x2 - 2x + 1


Multiplication of Polynomials

1.x - 3y and x - 7y
Answer: x2 - 10xy + 21y2
2.4a - 7b and 4a - 7b
Answer: 16a2 - 56ab + 49b2
3.6a + b and 6a + b
Answer: 36a2 + 12ab + b2
4.8x + y and 9x + 4y
Answer: 72x2 + 41xy + 4y2
5.2x - 7y and 2x + 7y
Answer: 4x2 - 49y2
6.3a - 4b and 9a + 2b
Answer: 27a2 - 30ab - 8b2
7.6mx + ny and 8mx - ny
Answer: 48m2x2 + 2mxny - n2y2
8.7ax + by and 8ax + 7by
Answer: 56a2x2 + 57axby + 7b2y2
9.(x - 5y - 6)2
Answer: x2 - 10xy + 25y2 - 12x + 60y + 36
10.(8p - 7q + 1)2
Answer: 64p2 - 112pq + 49q2 + 16p - 14q + 1
11.(8p - 7q + 4)2
Answer: 64p2 - 112pq + 49q2 + 64p - 56q + 16
12.(a - 5b - 6y)2
Answer: a2 - 10ab + 25b2 - 12a + 60b + 36y2
13.(5x4)(x + y)
Answer: 5x5 + 5x4y
14.(4x)(7x4 - 8y2)
Answer: 28x5 - 32xy2
15.(20xy)(3x4 - 8)
Answer: 60x5y - 160xy
16.(3ab4)(3a4 + 7)
Answer: 9a5b4 + 21ab4
17.(14s3t2)(3s3 - 7u2 + 7)
Answer: 42s6t2 - 98s3t2u2 + 98s3t2
18.(s + t)(u3 + v4)
Answer: su3 + sv4 + tu3 + tv4
19.(w + 4x)(5y + z)
Answer: 5wy + wz + 20xy + 4xz
20.(5s + 7t)(7u - 4v)
Answer: 35su - 20sv + 49tu - 28tv
21.(3w3 + 8x4)(2y2 + z3)
Answer: 6w3y2 + 3w3z3 + 16x4y2 + 8x4z3
22.(3w + 4x2)(4y2 - 3z4)
Answer: 12wy2 - 9wz4 + 16x2y2 - 12x2z4
23.(7a + 9b)(2b + 3)
Answer: 14ab + 21a + 18b2 + 27b
24.(x + 9y)(7y - 4)
Answer: 7xy - 4x + 63y2 - 36y
25.(8s3 + t)(4t4 - 7)
Answer: 32s3t4 - 56s3 + 4t5 - 7t
26.(s - t)(u + 8v)
Answer: su + 8sv - tu - 8tv
27.(x + 6)(x + 7)
Answer: x2 + 13x + 42
28.(x - 7)(x + 4)
Answer: x2 - 3x - 28
29.(x - 9)(x - 1)
Answer: x2 - 10x + 9
30.(8x + 1)(3x + 2)
Answer: 24x2 + 19x + 2
31.(x - 5)(4x + 3)
Answer: 4x2 - 17x - 15
32.(9x - 1)(5x - 2)
Answer: 45x2 - 23x + 2
33.(x + 9)(x + 9)
Answer: x2 + 18x + 81
34.(3x + 5)(3x + 5)
Answer: 9x2 + 30x + 25
35.(x - 1)(x - 1)
Answer: x2 - 2x + 1
36.(9x - 1)(9x - 1)
Answer: 81x2 - 18x + 1
37.[ 9x + 8(y - z) ] [ (9x - 8(y - z) ]
Answer: 81x2 - 64(y - z)2
38.(x + 2y) (x - 2y) (x2 + 4y2)
Answer: x4 - 16y4
39.(x - 8) (x2 + 8x + 64)
Answer: x3 - 512
40.(x2 - 7) (x4 + 7x2 + 49)
Answer: x6 - 343
41.(7y - 8) (49y2 + 56y + 64)
Answer: 343y3 - 512
42.(2y2 - 9) (4y4 + 18y2 + 81)
Answer: 8y6 - 729
43.(3a - 4b) (9a2 + 12ab + 16b2)
Answer: 27a3 - 64b3
44.(3a2 - 7b) (9a4 + 21a2b + 49b2)
Answer: 27a6 - 343b3
45.(9xy2 - 4z) (81x2y4 + 36xy2z + 16z2)
Answer: 729x3y6 - 64z3
46.x (2s + 9t)(4s2 - 18st + 81t2)
Answer: 8s3x + 729t3x
47.x (8s2 + 2t2)(64s4 - 16s2t2 + 4t4)
Answer: 512s6x + 8t6x
48.(8x + 5y) (64x2 - 40xy + 25y2)
Answer: 512x3 + 125y3
49.(7x2 + 9y2) (49x4 - 63x2y2 + 81y4)
Answer: 343x6 + 729y6
50.4x9 (y + 7z) (y - 7z)
Answer: 4x9y2 - 196x9z2


Division of Polynomials

1.63a4b7x9 ÷ 9a3b5x4
Answer: 7ab2x5
2.9x7y6z4 ÷ x2y3z3
Answer: 9x5y3z
3.-14m4n7 ÷ 2m3n4
Answer: -7mn3
4.72x2y2 ÷ 8xy
Answer: 9xy
5.24a6b5c7 ÷ 3a2b4c3
Answer: 8a4bc4
6.28x5y2z + 63x3y3z + 49x2y3z ÷ 7x2y2z
Answer: 4x3 + 9xy + 7y
7.x2 - 8x - 209 ÷ x - 19
Answer: x + 11
8.x2 - 29x + 204 ÷ x - 9
Answer: x - 20; remainder = 24
9.x2 - 22x + 88 ÷ x - 4
Answer: x - 18; remainder = 16
10.x4 + 3x2 - 460 ÷ x2 + 23
Answer: x2 - 20
11.x4 - 21x2 + 130 ÷ x2 - 10
Answer: x2 - 11; remainder = 20
12.x4 - 6x3y - 25x2y2 + 52xy3 + 14y4 ÷ x2 + 2xy - 7y2
Answer: x2 - 8xy - 2y2
13.x4 - 8x3y + 7x2y2 + 34xy3 + 15y4 ÷ x2 - 5xy - 3y2
Answer: x2 - 3xy - 5y2
14.8x3 - 57x2 + 48x + 42 ÷ x - 6
Answer: 8x2 - 9x - 6; remainder = 6
15.56x5 - 111x4 + 94x3 - 70x2 + 27x - 47 ÷ 7x3 - 6x2 - 2x - 5
Answer: 8x2 - 9x + 8; remainder = -2x - 7
16.6a5x4y9 ÷ a3x6y6
Answer: 6a2x-2y3
17.56p9q2r4 ÷ 8p3q6r5
Answer: 7p6q-4r-1
18.63x5 + 88x4 + 41x3 - 63x - 63x2 + 9 ÷ 7x3 + 9x2 + 9x - 1
Answer: 9x2 + x - 7; remainder = x + 2
19.32x3 + 44x2 + 25x + 21 ÷ 8x + 9
Answer: 4x2 + x + 2; remainder = 3
20.x4 + 11x3y + 25x2y2 - 7xy3 - 6y4 ÷ x2 + 8xy + 3y2
Answer: x2 + 3xy - 2y2
Posted by Gerardo Emerzona I

March 19, 2010

Algebra: Factoring of Fractional Polynomials

The following problems consist of three sets as described below.

  • Set 1 problems are reduction of fractions to their lowest terms.
  • Set 2 problems are multiplication of fractions.
  • Set 3 problems are division of fractions.

Most of the expressions are either products of a monomial and a binomial or products of two binomials.

Set 1

Reduction of fractional polynomials to their lowest terms.

Set 2

Multiplication of fractional polynomials.

Set 3

Division of fractional polynomials.

Posted by Gerardo Emerzona I

February 10, 2010

Algebra: Factoring of Polynomials

For convenience and quick browsing, the problems are grouped into polynomials that are:
  • product of a monomial and a polynomial
  • grouped into terms containing a common factor
  • difference of two squares
  • difference of two cubes
  • sum of two cubes
  • product of two binomials
The last group of polynomials given above consists of polynomials that are:
  • product of two binomials having a common factor
  • product of any two binomials
  • square of a binomial sum
  • square of a binomial difference
Some of these grouped polynomials contain factors which are a combination of a monomial, a binomial, and a trinomial. Polynomials that are:
  • difference of two squares can be the products of two binomials, or products of two trinomials, or products of a monomial and two binomials.
  • sum of two cubes can be products of a binomial and a trinomial, or products of a monomial, a binomial, and a trinomial

Polynomials With Monomial As One of Its Factors

1.15x4 + 15x3y
Answer: (3x3)(5x + 5y)
2.4x3 - 4x2y4
Answer: (4x2)(x - y4)
3.5x5 - 5x3y2
Answer: (x3)(5x2 - 5y2)
4.63s6t3 - 42s3t3u + 126s3t3
Answer: (21s3t3)(3s3 - 2u + 6)
5.14x5 + 6x2y
Answer: (2x2)(7x3 + 3y)
6.18x6y3 - 15x2y3z2 + 3x2y3
Answer: (3x2y3)(6x4 - 5z2 + 1)
7.14a4b4 + 56a3b4
Answer: (14a3b4)(a + 4)
8.7x5 + xy2
Answer: (x)(7x4 + y2)
9.7x4 + 21x3y3
Answer: (7x3)(x + 3y3)
10.24x6y4 - 192x3y4
Answer: (24x3y4)(x3 - 8)
11.35x3 - 49xy2
Answer: (7x)(5x2 - 7y2)
12.48a4b2 + 12a3b2
Answer: (12a3b2)(4a + 1)
13.56x4 - 40xy4
Answer: (8x)(7x3 - 5y4)
14.20a3b2 + 60ab2
Answer: (20ab2)(a2 + 3)
15.120s5t3 - 75s2t3u2 + 30s2t3
Answer: (15s2t3)(8s3 - 5u2 + 2)
16.24x4y - 6x3y
Answer: (6x3y)(4x - 1)
17.105x5y - 35x3y
Answer: (35x3y)(3x2 - 1)
18.30x5 - 15x2y3
Answer: (5x2)(6x3 - 3y3)
19.6x4 + 8x3y4
Answer: (x3)(6x + 8y4)
20.20x4y3 - 40xy3
Answer: (20xy3)(x3 - 2)

Polynomials That Can Be Grouped Into Terms Having A Common Factor

1.su4 + sv + tu4 + tv
Answer: (s + t)(u4 + v)
2.wy + 3wz + xy + 3xz
Answer: (w + x)(y + 3z)
3.24su - 42sv + 20tu - 35tv
Answer: (6s + 5t)(4u - 7v)
4.20w2y4 + 35w2z4 + 8xy4 + 14xz4
Answer: (5w2 + 2x)(4y4 + 7z4)
5.64w4y3 - 40w4z2 + 8xy3 - 5xz2
Answer: (8w4 + x)(8y3 - 5z2)
6.3ab + 3a + b2 + b
Answer: (3a + b)(b + 1)
7.5xy - 20x + 2y2 - 8y
Answer: (5x + 2y)(y - 4)
8.28s3t - 35s3 + 24t2 - 30t
Answer: (7s3 + 6t)(4t - 5)
9.7su + sv - 7tu - tv
Answer: (s - t)(7u + v)
10.su3 + sv4 + tu3 + tv4
Answer: (s + t)(u3 + v4)
11.8wy + 32wz + 3xy + 12xz
Answer: (8w + 3x)(y + 4z)
12.6su - 16sv + 3tu - 8tv
Answer: (2s + t)(3u - 8v)
13.12w2y + 15w2z2 + 8x3y + 10x3z2
Answer: (3w2 + 2x3)(4y + 5z2)
14.3w3y4 - 8w3z3 + 15x4y4 - 40x4z3
Answer: (w3 + 5x4)(3y4 - 8z3)
15.4ab + 2a + 10b2 + 5b
Answer: (2a + 5b)(2b + 1)
16.28xy - 4x + 7y2 - y
Answer: (4x + y)(7y - 1)
17.24st2 - 15s + 64t4 - 40t2
Answer: (3s + 8t2)(8t2 - 5)
18.5su + 2sv - 5tu - 2tv
Answer: (s - t)(5u + 2v)
19.12ab + 42a + 14b2 + 49b
Answer: (6a + 7b)(2b + 7)
20.6xy - 10x + 9y2 - 15y
Answer: (2x + 3y)(3y - 5)

Polynomials That Are The Difference of Two Squares

You will find that some of the problems given below can be factored twice, that is, after factoring an expression one of the resulting factors is still factorable.

Also, in some problems a group of terms is treated as a single term.

1.x2 - 25
Answer: (x - 5)(x + 5)
2.36x2 - 25
Answer: (6x - 5)(6x + 5)
3.25x2 - 49
Answer: (5x - 7)(5x + 7)
4.x2 - 9
Answer: (x - 3)(x + 3)
5.16x2 - 25
Answer: (4x - 5)(4x + 5)
6.x2 - (y - z)2
Answer: [ x + (y - z) ] [ (x - (y - z) ]
7.(x + 8)2 - 9
Answer: [ (x + 8) + 3 ] [ (x + 8) - 3 ]
8.(x + 3)2 - 1
Answer: [ (x + 3) + 1 ] [ (x + 3) - 1 ]
9.49x2 - 9(y - z)2
Answer: [ 7x + 3(y - z) ] [ (7x - 3(y - z) ]
10.(x + 5)2 - 9
Answer: [ (x + 5) + 3 ] [ (x + 5) - 3 ]
11.(x + 1)2 - 36
Answer: [ (x + 1) + 6 ] [ (x + 1) - 6 ]
12.(x + 8)2 - 25
Answer: [ (x + 8) + 5 ] [ (x + 8) - 5 ]
13.16x2 - 49(y - z)2
Answer: [ 4x + 7(y - z) ] [ (4x - 7(y - z) ]
14.49x4y2 - 49x4z2
Answer: 49x4 (y + z) (y - z)
15.49x2y2 - 441x2z2
Answer: 49x2 (y + 3z) (y - 3z)
16.9x2y2 - 36x2z2
Answer: 9x2 (y + 2z) (y - 2z)
17.16x2y2 - 144x2z2
Answer: 16x2 (y + 3z) (y - 3z)
18.4x2y2 - 100x2z2
Answer: 4x2 (y + 5z) (y - 5z)
19.x4 - 256y4
Answer: (x + 4y) (x - 4y) (x2 + 16y2)
20.x4 - 81y4
Answer: (x + 3y) (x - 3y) (x2 + 9y2)

Polynomials That Are The Difference of Two Cubes

1.27x3y6 - 125z3
Answer: (3xy2 - 5z) (9x2y4 + 15xy2z + 25z2)
2.343y6 - 512
Answer: (7y2 - 8) (49y4 + 56y2 + 64)
3.8a6 - 125b3
Answer: (2a2 - 5b) (4a4 + 10a2b + 25b2)
4.x3 - 8
Answer: (x - 2) (x2 + 2x + 4)
5.8a3 - 512b3
Answer: (2a - 8b) (4a2 + 16ab + 64b2)
6.343x3y6 - 64z3
Answer: (7xy2 - 4z) (49x2y4 + 28xy2z + 16z2)
7.512y3 - 125
Answer: (8y - 5) (64y2 + 40y + 25)
8.27a6 - 125b3
Answer: (3a2 - 5b) (9a4 + 15a2b + 25b2)
9.8a3 - 125b3
Answer: (2a - 5b) (4a2 + 10ab + 25b2)
10.125y6 - 216
Answer: (5y2 - 6) (25y4 + 30y2 + 36)
11.343y3 - 216
Answer: (7y - 6) (49y2 + 42y + 36)
12.x6 - 8
Answer: (x2 - 2) (x4 + 2x2 + 4)
13.8y3 - 125
Answer: (2y - 5) (4y2 + 10y + 25)
14.8y3 - 216
Answer: (2y - 6) (4y2 + 12y + 36)
15.343y6 - 27
Answer: (7y2 - 3) (49y4 + 21y2 + 9)
16.x6 - 64
Answer: (x2 - 4) (x4 + 4x2 + 16)
17.x6 - 27
Answer: (x2 - 3) (x4 + 3x2 + 9)
18.x3 - 27
Answer: (x - 3) (x2 + 3x + 9)
19.8y6 - 343
Answer: (2y2 - 7) (4y4 + 14y2 + 49)
20.27y6 - 27
Answer: (3y2 - 3) (9y4 + 9y2 + 9)
21.512a3 - 8b3
Answer: (8a - 2b) (64a2 + 16ab + 4b2)
22.64a6 - 343b3
Answer: (4a2 - 7b) (16a4 + 28a2b + 49b2)
23.8a3 - 8b3
Answer: (2a - 2b) (4a2 + 4ab + 4b2)
24.x3 - 125
Answer: (x - 5) (x2 + 5x + 25)
25.512a6 - 8b3
Answer: (8a2 - 2b) (64a4 + 16a2b + 4b2)

Polynomials That Are The Sum of Two Cubes

1.512x6 + 27y6
Answer: (8x2 + 3y2) (64x4 - 24x2y2 + 9y4)
2.8x6 + 27y6
Answer: (2x2 + 3y2) (4x4 - 6x2y2 + 9y4)
3.8a6 + 125b6
Answer: (2a2 + 5b2) (4a4 - 10a2b2 + 25b4)
4.512x3 + 8y3
Answer: (8x + 2y) (64x2 - 16xy + 4y2)
5.8x6 + 64y6
Answer: (2x2 + 4y2) (4x4 - 8x2y2 + 16y4)
6.8m3 + 343n3
Answer: (2m + 7n) (4m2 - 14mn + 49n2)
7.64s3x + 8t3x
Answer: x (4s + 2t)(16s2 - 8st + 4t2)
8.8x3 + 216y3
Answer: (2x + 6y) (4x2 - 12xy + 36y2)
9.125x6y + 343y
Answer: y (5x2 + 7)(25x4 - 35x2 + 49)
10.125x3 + 8y3
Answer: (5x + 2y) (25x2 - 10xy + 4y2)
11.512x6 + 343y6
Answer: (8x2 + 7y2) (64x4 - 56x2y2 + 49y4)
12.8x3 + 125y3
Answer: (2x + 5y) (4x2 - 10xy + 25y2)
13.125a6x + 64b6x
Answer: x (5a2 + 4b2)(25a4 - 20a2b2 + 16b4)
14.64m3y + 27n3y
Answer: y (4m + 3n)(16m2 - 12mn + 9n2)
15.64s3x + 125t3x
Answer: x (4s + 5t)(16s2 - 20st + 25t2)
16.216x3z + 343y3z
Answer: z (6x + 7y)(36x2 - 42xy + 49y2)
17.8s6x + 125t6x
Answer: x (2s2 + 5t2)(4s4 - 10s2t2 + 25t4)
18.343a3c + 64b3c
Answer: c (7a + 4b)(49a2 - 28ab + 16b2)
19.512s6x + 343t6x
Answer: x (8s2 + 7t2)(64s4 - 56s2t2 + 49t4)
20.8s6u + 27t6u
Answer: u (2s2 + 3t2)(4s4 - 6s2t2 + 9t4)


The following problems are about polynomials which are the products of two binomials.

Polynomials That Are The Products of Two Binomials Having a Common Factor

1.x2 + 11x + 30
Answer: (x + 5)(x + 6)
2.x2 - 11x + 24
Answer: (x - 3)(x - 8)
3.x2 + 8x + 15
Answer: (x + 3)(x + 5)
4.49x2 - 35x + 4
Answer: (7x - 1)(7x - 4)
5.x2 - 9x + 14
Answer: (x - 2)(x - 7)
6.x2 + 5x + 4
Answer: (x + 1)(x + 4)
7.x2 - x - 30
Answer: (x - 6)(x + 5)
8.x2 + 7x + 10
Answer: (x + 5)(x + 2)
9.25x2 + 65x + 42
Answer: (5x + 6)(5x + 7)
10.x2 + x - 56
Answer: (x - 7)(x + 8)
11.x2 - 9x + 8
Answer: (x - 1)(x - 8)
12.x2 + 2x - 15
Answer: (x - 3)(x + 5)
13.x2 + 13x + 42
Answer: (x + 6)(x + 7)
14.x2 + 11x + 24
Answer: (x + 8)(x + 3)
15.x2 - 11x + 30
Answer: (x - 5)(x - 6)
16.x2 - 13x + 40
Answer: (x - 5)(x - 8)
17.x2 + x - 30
Answer: (x - 5)(x + 6)
18.x2 + 3x + 2
Answer: (x + 2)(x + 1)
19.x2 - 2x - 3
Answer: (x - 3)(x + 1)
20.x2 - 8x + 15
Answer: (x - 5)(x - 3)

Polynomials That Are The Products of Any Two Binomials

1.24x2 + 35x - 49
Answer: (8x - 7)(3x + 7)
2.2x2 + 7x - 4
Answer: (2x - 1)(x + 4)
3.24x2 + 34x + 7
Answer: (6x + 7)(4x + 1)
4.32x2 + 28x + 3
Answer: (8x + 1)(4x + 3)
5.33x2 + 14x + 1
Answer: (3x + 1)(11x + 1)
6.12x2 + 11x - 5
Answer: (3x - 1)(4x + 5)
7.5x2 + 33x - 56
Answer: (5x - 7)(x + 8)
8.8x2 - 9x + 1
Answer: (x - 1)(8x - 1)
9.3x2 + 19x + 6
Answer: (x + 6)(3x + 1)
10.6x2 - 25x + 14
Answer: (3x - 2)(2x - 7)
11.35x2 + 17x - 30
Answer: (7x - 5)(5x + 6)
12.7x2 + 38x + 15
Answer: (7x + 3)(x + 5)
13.3x2 + 32x + 64
Answer: (x + 8)(3x + 8)
14.3x2 - 13x + 12
Answer: (x - 3)(3x - 4)
15.10x2 + 21x + 9
Answer: (2x + 3)(5x + 3)
16.6x2 - 23x + 20
Answer: (3x - 4)(2x - 5)
17.10x2 - 47x + 42
Answer: (2x - 7)(5x - 6)
18.6x2 - 43x + 42
Answer: (6x - 7)(x - 6)
19.15x2 + 4x - 32
Answer: (3x - 4)(5x + 8)
20.3x2 - 14x - 5
Answer: (x - 5)(3x + 1)

Polynomials That Are The Square of a Binomial Sum

1.169x2 + 364x + 196
Answer: (13x + 14)(13x + 14)
2.64x2 + 80x + 25
Answer: (8x + 5)(8x + 5)
3.25x2 + 40x + 16
Answer: (5x + 4)(5x + 4)
4.x2 + 10x + 25
Answer: (x + 5)(x + 5)
5.64x2 + 48x + 9
Answer: (8x + 3)(8x + 3)
6.25x2 + 60x + 36
Answer: (5x + 6)(5x + 6)
7.16x2 + 8x + 1
Answer: (4x + 1)(4x + 1)
8.x2 + 22x + 121
Answer: (x + 11)(x + 11)
9.169x2 + 78x + 9
Answer: (13x + 3)(13x + 3)
10.x2 + 4x + 4
Answer: (x + 2)(x + 2)
11.49x2 + 56x + 16
Answer: (7x + 4)(7x + 4)
12.x2 + 2x + 1
Answer: (x + 1)(x + 1)
13.x2 + 14x + 49
Answer: (x + 7)(x + 7)
14.169x2 + 156x + 36
Answer: (13x + 6)(13x + 6)
15.x2 + 16x + 64
Answer: (x + 8)(x + 8)
16.x2 + 26x + 169
Answer: (x + 13)(x + 13)
17.64x2 + 16x + 1
Answer: (8x + 1)(8x + 1)
18.x2 + 8x + 16
Answer: (x + 4)(x + 4)
19.25x2 + 110x + 121
Answer: (5x + 11)(5x + 11)
20.49x2 + 42x + 9
Answer: (7x + 3)(7x + 3)

Polynomials That Are The Square of a Binomial Difference

1.9x2 - 30x + 25
Answer: (3x - 5)(3x - 5)
2.169x2 - 364x + 196
Answer: (13x - 14)(13x - 14)
3.25x2 - 40x + 16
Answer: (5x - 4)(5x - 4)
4.49x2 - 42x + 9
Answer: (7x - 3)(7x - 3)
5.9x2 - 60x + 100
Answer: (3x - 10)(3x - 10)
6.64x2 - 80x + 25
Answer: (8x - 5)(8x - 5)
7.36x2 - 84x + 49
Answer: (6x - 7)(6x - 7)
8.x2 - 26x + 169
Answer: (x - 13)(x - 13)
9.9x2 - 12x + 4
Answer: (3x - 2)(3x - 2)
10.x2 - 2x + 1
Answer: (x - 1)(x - 1)
11.x2 - 14x + 49
Answer: (x - 7)(x - 7)
12.x2 - 4x + 4
Answer: (x - 2)(x - 2)
13.4x2 - 20x + 25
Answer: (2x - 5)(2x - 5)
14.x2 - 8x + 16
Answer: (x - 4)(x - 4)
15.25x2 - 80x + 64
Answer: (5x - 8)(5x - 8)
16.x2 - 16x + 64
Answer: (x - 8)(x - 8)
17.x2 - 6x + 9
Answer: (x - 3)(x - 3)
18.64x2 - 16x + 1
Answer: (8x - 1)(8x - 1)
19.x2 - 10x + 25
Answer: (x - 5)(x - 5)
20.49x2 - 70x + 25
Answer: (7x - 5)(7x - 5)
Posted by Gerardo Emerzona I
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