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The distributive law

Algebra · level 5, lesson 1 of 6

A question to start

Work out \(7 \times 98\) in your head, without paper.

Most people freeze, because 98 is an awkward number. But 98 is just \(100 - 2\). Seven hundreds is 700, and seven twos is 14, so

\[7 \times 98 = 7 \times 100 - 7 \times 2 = 700 - 14 = 686.\]

Here is another one: \(23 \times 4 + 77 \times 4\). You could do two multiplications and an addition. Or you could notice that both parts are "something times 4", so the whole thing is \((23 + 77) \times 4 = 100 \times 4 = 400\).

Both tricks use the same law of arithmetic, the distributive law. It is the most used rule in all of algebra: it is how brackets are opened and closed, and almost every lesson after this one leans on it. This lesson shows what it says, why it is true (with a picture), and how to use it forwards and backwards.

The idea

What the law says

For any numbers \(a\), \(b\) and \(c\):

\[a(b + c) = ab + ac.\]

In words: multiplying a sum by a number is the same as multiplying each part of the sum by that number and then adding. The number outside the bracket is "distributed" (handed out) to every term inside, like handing one leaflet to every person in a room.

The same is true for a difference: \(a(b - c) = ab - ac\). And it works for any number of terms inside: \(a(b + c + d) = ab + ac + ad\).

Two words you will use all the time:

  • Expanding means using the law to remove brackets: \(3(x + 4) \to 3x + 12\).
  • Factoring (or factorising) means using it backwards to put brackets in: \(3x + 12 \to 3(x + 4)\).

Why it is true: the area picture

Think of \(3(x + 4)\) as the area of a rectangle that is 3 tall and \(x + 4\) long. The long side is made of two pieces, one of length \(x\) and one of length 4. Draw a line where the two pieces meet, and the rectangle splits into two smaller rectangles.

x 4 3 3x 12 whole area 3(x + 4) 3(x + 4) = 3x + 12

The whole rectangle has area \(3(x + 4)\), "height times length". The two pieces have areas \(3 \times x = 3x\) and \(3 \times 4 = 12\). Cutting a rectangle into two does not change the total area, so

\[3(x + 4) = 3x + 12.\]

This is true whatever length \(x\) is, because the picture works for any length. That is the whole reason the law holds for positive numbers.

Why it is true: counting groups

Here is a second reason, with no picture. \(3(x + 4)\) means "3 lots of \((x + 4)\)":

\[3(x + 4) = (x + 4) + (x + 4) + (x + 4).\]

Now collect like terms (level 4): there are three \(x\)'s and three 4's. So the total is \(3x + 12\). Each copy of the bracket brings one \(x\) and one 4, so the 3 multiplies both.

Checking with a number

Any time you expand, you can test your answer by putting in a number. Take \(x = 5\):

  • \(3(x + 4) = 3 \times 9 = 27\);
  • \(3x + 12 = 15 + 12 = 27\). ✓

If the two sides disagree for even one number, the expansion is wrong. (If they agree, it is very likely right; try a second number if you want to be sure.)

Negative numbers outside the bracket

What is \(-2(x - 5)\)? The law still says: multiply every term inside by \(-2\).

\[-2(x - 5) = (-2) \times x + (-2) \times (-5) = -2x + 10.\]

The second term is \(+10\) because a negative times a negative is positive (see multiplying integers). Check with \(x = 1\): the left side is \(-2 \times (1 - 5) = -2 \times (-4) = 8\), and the right side is \(-2 + 10 = 8\). ✓

A plain minus sign in front of a bracket is the same as \(-1\) times the bracket. So

\[-(x - 3) = -x + 3.\]

Every sign inside the bracket flips. This is the place where most algebra mistakes happen, so slow down whenever you see a minus in front of a bracket.

Letters outside the bracket

The number outside can be a letter, or a mix of letters and numbers. You multiply each term in the usual way:

  • \(x(x + 3) = x \times x + x \times 3 = x^2 + 3x\);
  • \(2a(3a - b + 4) = 6a^2 - 2ab + 8a\).

The area picture still works: \(x(x + 3)\) is a rectangle \(x\) tall and \(x + 3\) long, made of an \(x\) by \(x\) square (area \(x^2\)) and an \(x\) by 3 rectangle (area \(3x\)).

Expanding and simplifying

Often there are several brackets. Expand each one, then collect like terms.

\[5(x + 2) - 2(x - 4) = 5x + 10 - 2x + 8 = 3x + 18.\]

Look at the \(+8\): it came from \((-2) \times (-4)\). The \(-2\) belongs to the second bracket, sign and all.

Factoring: the law backwards

Factoring starts with \(ab + ac\) and finds the \(a(b + c)\). You look for a common factor: a number (or letter) that divides every term.

Take \(6x + 15\). Both 6 and 15 are multiples of 3, so

\[6x + 15 = 3 \times 2x + 3 \times 5 = 3(2x + 5).\]

To factor fully, take out the greatest common factor (GCF): the biggest number and the most letters that go into every term. In \(12a - 8ab\) the greatest common factor of 12 and 8 is 4, and both terms contain \(a\), so

\[12a - 8ab = 4a(3 - 2b).\]

You could also write \(2(6a - 4ab)\), which is true but not fully factored: the bracket still has a common factor of \(2a\) inside it.

To find what goes in the bracket, divide each term by the common factor: \(12a \div 4a = 3\) and \(-8ab \div 4a = -2b\). Then always check by expanding: \(4a(3 - 2b) = 12a - 8ab\). ✓

Why factoring is useful

Expanding and factoring are opposite directions on the same road. Expanding makes things easier to add; factoring makes things easier to divide and to solve. You will see this in level 6, where factoring \(x^2 + 5x + 6\) into \((x + 2)(x + 3)\) is the key to solving equations with \(x^2\) in them. It also explains mental tricks like the ones at the start:

\[23 \times 4 + 77 \times 4 = 4(23 + 77) = 4 \times 100 = 400.\]

Dividing a sum

The distributive law also tells you how to divide a sum: each term gets divided.

\[\frac{6x + 9}{3} = \frac{6x}{3} + \frac{9}{3} = 2x + 3.\]

That is because dividing by 3 is multiplying by \(\tfrac13\), and \(\tfrac13\) distributes like any other number. The fraction bar acts like a bracket around the whole top.

Worked examples

1. A simple expansion. Expand \(4(2x + 3)\).

  • Multiply each term by 4: \(4 \times 2x = 8x\) and \(4 \times 3 = 12\).
  • So \(4(2x + 3) = 8x + 12\).
  • Check with \(x = 1\): \(4 \times 5 = 20\) and \(8 + 12 = 20\). ✓

2. A negative outside. Expand \(-3(2x - 5)\).

  • \((-3) \times 2x = -6x\).
  • \((-3) \times (-5) = +15\).
  • So \(-3(2x - 5) = -6x + 15\).
  • Check with \(x = 2\): \(-3 \times (4 - 5) = -3 \times (-1) = 3\), and \(-12 + 15 = 3\). ✓

3. Expand and simplify. Simplify \(5(x + 2) - 2(x - 4)\).

  • First bracket: \(5x + 10\).
  • Second bracket with its \(-2\): \(-2x + 8\).
  • Together: \(5x + 10 - 2x + 8 = 3x + 18\).
  • Check with \(x = 1\): \(5 \times 3 - 2 \times (-3) = 15 + 6 = 21\), and \(3 + 18 = 21\). ✓

4. Factoring. Factor \(18x - 24\) fully.

  • The greatest common factor of 18 and 24 is 6. The second term has no \(x\), so \(x\) is not a common factor.
  • Divide each term by 6: \(18x \div 6 = 3x\) and \(-24 \div 6 = -4\).
  • So \(18x - 24 = 6(3x - 4)\). Check: \(6 \times 3x - 6 \times 4 = 18x - 24\). ✓

5. Mental arithmetic. Work out \(6 \times 47\) in your head.

  • \(47 = 40 + 7\), so \(6 \times 47 = 6 \times 40 + 6 \times 7 = 240 + 42 = 282\).
  • This is exactly what a rectangle picture with sides 6 and \(40 + 7\) shows, and it is what written multiplication does too: it multiplies the tens and the ones separately and adds.

6. Letters outside and inside. Expand \(3x(x - 2y + 1)\).

  • \(3x \times x = 3x^2\); \(3x \times (-2y) = -6xy\); \(3x \times 1 = 3x\).
  • So \(3x(x - 2y + 1) = 3x^2 - 6xy + 3x\).

Try it

  1. Expand \(5(x + 3)\).
  2. Expand \(-4(y - 2)\).
  3. Expand and simplify \(3(2a + 1) + 2(a - 5)\).
  4. Expand and simplify \(7(x + 1) - 3(x - 2)\).
  5. Factor fully: (a) \(12x + 18\); (b) \(15ab - 10a\).
  6. Work out \(9 \times 999\) in your head, and say which law you used.
  7. Two rectangles are both 4 wide. One is \(x + 7\) long and the other is \(x - 2\) long. Write their total area as an expression, simplify it, then factor the answer.
  8. Sam says \(-(x - 6) = -x - 6\). Test his claim with \(x = 10\), then give the correct expansion.
Answers
  1. \(5x + 15\). Check with \(x = 1\): \(5 \times 4 = 20\) and \(5 + 15 = 20\). ✓
  2. \((-4) \times y = -4y\) and \((-4) \times (-2) = +8\), so \(-4y + 8\).
  3. \(6a + 3 + 2a - 10 = 8a - 7\). Check with \(a = 1\): \(3 \times 3 + 2 \times (-4) = 9 - 8 = 1\) and \(8 - 7 = 1\). ✓
  4. \(7x + 7 - 3x + 6 = 4x + 13\). (The \(+6\) is \((-3) \times (-2)\).) Check with \(x = 0\): \(7 + 6 = 13\). ✓
  5. (a) The greatest common factor of 12 and 18 is 6: \(12x + 18 = 6(2x + 3)\). (b) Both terms have the factor \(5a\): \(15ab \div 5a = 3b\) and \(10a \div 5a = 2\), so \(15ab - 10a = 5a(3b - 2)\).
  6. \(9 \times 999 = 9 \times (1000 - 1) = 9000 - 9 = 8991\), by the distributive law.
  7. Total area \(4(x + 7) + 4(x - 2) = 4x + 28 + 4x - 8 = 8x + 20\). Factored: \(8x + 20 = 4(2x + 5)\). That makes sense: put the two rectangles end to end and you get one rectangle 4 wide and \((x + 7) + (x - 2) = 2x + 5\) long.
  8. With \(x = 10\): \(-(10 - 6) = -4\), but \(-10 - 6 = -16\). They disagree, so Sam is wrong. The minus flips every sign: \(-(x - 6) = -x + 6\), and \(-10 + 6 = -4\). ✓

Common mistake

Multiplying only the first term. \(3(x + 4)\) is not \(3x + 4\). The 3 multiplies everything inside. The area picture shows it: the rectangle has two pieces, and both of them are 3 tall. Check with \(x = 1\): \(3 \times 5 = 15\), but \(3 + 4 = 7\).

Losing a sign. In \(10 - 2(x - 3)\) the \(-2\) multiplies the \(-3\) too: \(10 - 2x + 6 = 16 - 2x\). Writing \(10 - 2x - 6\) is wrong. When there is a minus in front of a bracket, write the multiplication of each term out in full until it feels automatic.

Where it leads