Learn – Algebra foundations – Multiplication and Factoring

1 Multiplication without distributive property

1.1 Exercises

1.1.1 Problem set

Simplify the following.

  1. 1\times x
  2. 2\times (-x)
  3. -x\times (-y)
  4. -3\times (-x)

1.1.2 Problem set

Simplify the following.

  1. 3\times 6x
  2. -2x\times 4
  3. 4x\times (-5)
  4. 7x\times (-2)
  5. -3x\times (-5)

1.1.3 Problem set

Simplify the following.

  1. x\times x
  2. 1\times x\times x
  3. 1\times x\times (-x)
  4. 2\times x\times x \times y
  5. 1\times x\times y \times y\times x

1.1.4 Problem set

Simplify the following.

  1. (-a)\times (-a)
  2. 2\times (-x)\times (-x)
  3. 1\times x\times x \times x
  4. (-a)\times(-b)\times(-c)
  5. 4\times (-x)\times (-x) \times (-x)
  6. -(-a)
  7. 3x\times (-y)\times (-x) \times (-4)

1.1.5 Problem set

Simplify the following.

  1. 2\times x\times y\times (-x)\times(-y)
  2. -3\times (-a)\times b\times (-b)\times(a)
  3. -5\times p\times (-q)\times p\times (-p)\times (-q)
  4. -x\times (-2y)\times 3x\times y
  5. -10a\times (-a)\times 2b\times (-2a)\times (-b)\times (-2b)
  6. -2\times (-x)\times (-2y)\times x\times (-2y)\times (-2x)
  7. -3\times (-p)\times (-2q)\times 2r\times (-2r)\times 2p

1.1.6 Problem set

Simplify the following.

  1. 2\times 3x^2
  2. -3x^2\times 4
  3. 2xy\times 4xy
  4. -3ab^2 \times (-3a^2b)
  5. -5pq \times 2pq^2

2 Factoring M expression forms

2.1 Exercises

2.1.1 Problem set

For each of the following, write as a product of factors and circle the factors.

  1. xyz
  2. 2x
  3. -3x
  4. 6xy
  5. 9x^2
  6. 10x^2y

2.1.2 Problem set

For each of the following, write as a product of factors and circle the factors.

  1. x^3
  2. a^4
  3. x^3y^2
  4. p^2q^3
  5. x^2y^2z^3
  6. ab^2c^3
  7. pq^2p^2
  8. xy^2zw^2

2.1.3 Problem set

For each of the following, write as a product of factors and circle the factors.

  1. x+2
  2. a+b
  3. x^2+3
  4. pq+6
  5. xy+zw

2.1.4 Problem set

For each of the following, write as a product of factors and circle the factors.

  1. x(x-1)
  2. (p+3)x
  3. (a+1)(b+1)
  4. (x+4)^2
  5. (p-7)^3
  6. (a+2)^3(b+3)

2.1.5 Problem set

For each of the following, write as a product of factors and circle the factors.

  1. x(x^2+11)
  2. x^2(x+5)
  3. (x+1)(x+2)(x+3)
  4. (2x+1)(3x+2)(4x+3)
  5. (x-1)(x+1)(2x+1)(2x-1)
  6. (x-2)(x+2)(x^2+2)
  7. (2y-1)^2(y+1)
  8. (x+1)^2(x^2+1)
  9. (x+1)^2(x-1)^2

2.1.6 Problem set

For each of the following, write as a product of factors and circle the factors.

  1. yz+4
  2. x^2+yz
  3. (x+1)(x+2)+6
  4. (x+3)^2+6
  5. (x-5)^3+10
  6. (2x+1)(3x+2)(4x+3)+5
  7. (2x+1)(3x+2)(4x+3)+(5x+4)(6x+5)(7x+6)
  8. x^3+2x+12
  9. x^4+3x^2-4x-12

3 Multiplication with distributive property

3.1 Exercises

3.1.1 Problem set

For each of the following expressions, multiply the parentheses out.

  1. x(2x-1)
  2. x^2(2x+x^2)
  3. p(p-q)

3.1.2 Problem set

For each of the following expressions, multiply the parentheses out.

  1. (2x+3)(x+5)
  2. (3x+5)(4x+2)
  3. (-x-1)(2x+5)
  4. (2x-1)(-2x+1)
  5. (-3x+2)(-5x-3)

3.1.3 Problem set

For each of the following expressions, multiply the parentheses out.

  1. (-4x+3)(-3x+5)
  2. (-7x-3)(-2x+3)
  3. (x^2-3x+1)(2x-3)
  4. (2x^2-1)(3x^2-5x-1)
  5. (3x^2-2x-1)(3x^2-2x-1)

3.1.4 Problem set

Get rid of parentheses in the following by multiplying as appropriate.

  1. (x^2 + 3)(2x)
  2. (2x^2 - 1)(2x - 1)
  3. ((x+1)(x+2))(2x - 1)
  4. \left(\frac{x+1}{x+2}\right)(2x - 1)
  5. ((x+1)(x+2)+x+3)(2x - 1)
  6. \left(\frac{x+1}{x+2}+x+3\right)(2x - 1)

4 Factoring A/S expression forms

4.1 Exercises

4.1.1 Problem set

For each of the following expressions, write as a product of factors.

  1. 6x+8
  2. 21x-14
  3. 9x-15
  4. 9x+3
  5. 15x-5
  6. 12x-8
  7. 6\times 4\times x-8

4.1.2 Problem set

For each of the following expressions, write as a product of factors.

  1. 3x^2-2x
  2. 5p-7p^2
  3. 4a^2-3ab^2
  4. 8p^2q^2-6pq^3
  5. 6x^3-x
  6. a^4+a^2+a
  7. 6x^3-3x
  8. p^9+2p^7+p

4.1.3 Problem set

For each of the following expressions, write as a product of factors.

  1. p^5+2pq+p
  2. x^2(x^3+x)
  3. a^2(ab+ac)
  4. a^{100}-a^{50}
  5. a^{150}-a^{100}+a^{50}

4.1.4 Problem set

For each of the following expressions, write as a product of factors, with one factor being a negative number.

  1. -15x-10
  2. -16x-12
  3. -8x-8
  4. -3\times 2\times x-4
  5. -5\times 7\times x-7
  6. -10\times 9\times x-6

4.1.5 Problem set

For each of the following expressions, write as a product of factors, with one factor being a negative number.

  1. -15x+9
  2. -24x+16
  3. -12x+12
  4. -9\times 7\times x+14
  5. -5\times 8\times x+8
  6. -9\times 14\times x+21

4.1.6 Problem set

For each of the following expressions, write as a product of factors.

  1. a(a+2)+2a
  2. 5a(a+2)+2(a+2)
  3. (a-1)(a+2)+2(a-1)

4.1.7 Problem set

For each of the following expressions, write as a product of factors.

  1. (p+1)(3p+2)+2(3p+2)
  2. 3(3p+2)-(1-p)(3p+2)
  3. (a-1)^2+3(a-1)
  4. (2p-1)^2-(2p-1)
  5. (3m-1)^2-(m-1)(3m-1)

5 Miscellaneous

5.1 Exercises

5.1.1 Problem set

Simplify the following.

  1. (\sqrt{5} + 5)(1 - \sqrt{5})
  2. (\sqrt{5} - 5)(1 + \sqrt{5})
  3. (\sqrt{2} + \sqrt{3})\sqrt{6}. (Express in terms of \sqrt{2} and \sqrt{3})
  4. \frac{\sqrt{5} + 25}{\sqrt{5}}
  5. \frac{\sqrt{3}\sqrt{6} + 2}{\sqrt{2}}

5.1.2 Problem set

Simplify the following.

  1. Simplify: a(bc)(de) - e(db)(ca)
  2. abc+dba = 2 and ab = 2. What is d+c?
  3. 19 + b = c + 21. What is 3b - 3c?
  4. ab = \frac{1}{2}, c = d + 2. What is the value of abc - abd?