Learn – Algebra foundations – Distributive property

1 Distributing factors

1.1 Exercises

1.1.1 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. a \times (b + c)
  2. a \times (b - c)
  3. (p + q)\times r
  4. (x - y)\times z
  5. a\times (-b - c)

1.1.2 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. (-b - c) a
  2. (x + y)(-z)
  3. (-p + q)(-r)
  4. (a - b)(-c)
  5. (-x - y)(-z)

1.1.3 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. a(a + b)
  2. (-p)(p - q)
  3. (-x)(-y+x)
  4. (-a)(-b-a)
  5. -p(-p+q)

1.1.4 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

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

1.1.5 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. x(x^2+y)
  2. x\left((x+1)x\right)
  3. \left(xy+x+y\right)xy
  4. \left((a+1)a\right)+b
  5. \left(x^3-x(x+2)\right)x
  6. \left(-p^3-p(p-2)\right)p

1.1.6 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

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

1.1.7 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. \frac{x(y+1)-x^2}{x}
  2. \frac{x(y+1)-x^2y}{y}
  3. \frac{x(y+1)+(x+1)y}{xy}
  4. \frac{x(y+1)+(x+1)y}{x(y+1)}
  5. \frac{x(y+1)+(x+1)y}{(x+1)(y+1)}

2 Distributing signs

2.1 Exercises

2.1.1 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. +(a+b)
  2. +(p-q)
  3. +(-x+y)
  4. +(-a-b+c-d)

2.1.2 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. -(p+q)
  2. -(x-y)
  3. -(-a+b)
  4. -(-p-q)
  5. -(-x-y+z+w-u-v)

2.1.3 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. a-(b+c)
  2. x-(y+z)
  3. p-(q-r)
  4. a-(b-c)
  5. x-(-y+z)
  6. -p-(-q+r)
  7. a-(-b-c)
  8. -x-(-y-z)

2.1.4 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. (a+b)-c
  2. (x-y)+z
  3. (p-q)-r
  4. (-a+b)-c
  5. (-x-y)+z
  6. (-p-q)-r

2.1.5 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. -(a+b)+c
  2. -(x-y)-z
  3. -(-p+q)-r
  4. -(-a-b)+c

3 When not to distribute

3.1 Exercises

3.1.1 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. a (b \times c)
  2. (b \times c \times d)a
  3. (xyz)(-a)
  4. (a\times b)(c\times d)

3.1.2 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. a\left(\frac{b}{c}\right)
  2. \left(\frac{b}{c}\right) a
  3. (ab)\left(\frac{x}{y}\right)

3.1.3 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. a+(b\times c)
  2. z-(x\times y)
  3. z+\left(\frac{x}{y}\right)
  4. (a\times c)+b

3.1.4 Problem set

For each of the following, get rid of parentheses appropriately and simplify.

  1. b - \left((a+1)a\right)
  2. b - \left(\frac{a+1}{a}\right)
  3. b \times \left(\frac{a+1}{a}\right)
  4. b \times \left(\frac{a+1}{a+2}\right)

4 Distributing from either side

4.1 Exercises

4.1.1 Problem set

Apply distributive property, if you can, to the following problems to get rid of parentheses.

  1. 2(a+b)3
  2. (x-y)-3
  3. -(a-b)3
  4. -(a-b)-3
  5. -(a-b)(-3)

4.1.2 Problem set

Apply distributive property, if you can, to the following problems to get rid of parentheses.

  1. -2(p-q)-3
  2. -2(a-b)(-3)
  3. -2(p-q-r)(-3)
  4. (p-q)-(q-p)
  5. (p-q)-(r)
  6. -3(a-b)2-3(b-a)
  7. -(a-b)3-3(b-a)

5 Splitting fractions

5.1 Exercises

5.1.1 Problem set

Write each of the following fractions, if possible, as a sum of two or more fractions.

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

5.1.2 Problem set

Write each of the following fractions, if possible, as a sum of two or more fractions.

  1. \frac{x+1}{x+3}
  2. \frac{x^2+1}{x^2}
  3. -\frac{xy-x}{x}
  4. -\frac{xy+x^2-x}{xy}
  5. -\frac{x^3-x}{2x^2}

5.1.3 Problem set

Write each of the following fractions, if possible, as a sum of two or more fractions.

  1. \frac{2x^2+5}{x^5+4}
  2. \frac{2-x^5}{2-x^4}
  3. \frac{9a^2-4a-100}{2a^2-3a+2}
  4. \frac{(p+4)^2+(p+2)^2}{(p+4)^2+2}
  5. \frac{(x-6)^2+(x-2)^2}{x-2}