Every term meets every term

The idea

(a + m)(b + n) = ab + mb + an + mn: the product is the sum of the products of each term of the first expression with each term of the second.

Both numbers grow

Take 23 × 27 = 621. Now the first number grows by 2 and the second by 3. It becomes 25 × 30 = 750.

The product grew by 129. Which parts add up to 129?

• 2 × 3
• 2 × 27 + 3 × 23
• 2 × 27 + 3 × 23 + 2 × 3

Fill the four cells

The rectangle is (a + m) wide and (b + n) high. Tap a cell to collect its product. Which terms meet there?

Each term times each term

(a + m)(b + n) = ab + mb + an + mn The product is the sum of the products of each term of the first bracket with each term of the second.

Two terms times two terms makes 2 × 2 = 4 products. The first one, ab, is the old product. The rest is the increase: an + bm + mn.

(23 + 2)(27 + 3) = 23 × 27 + 2 × 27 + 23 × 3 + 2 × 3 = 621 + 54 + 69 + 6 = 750

Notes

(a + m)(b + n) = ab + mb + an + mn Each term of the first bracket multiplies each term of the second, and the four products are added.

Check yourself

13 × 12 = 156. The first number grows by 2 and the second by 5, so it becomes 15 × 17. By how much does the product increase?

Answer: 99

an = 13 × 5 = 65, bm = 12 × 2 = 24 and mn = 2 × 5 = 10. Together 65 + 24 + 10 = 99. Check: 255 − 156 = 99.

Which is the expansion of (x + 4)(y + 5)?

Write 21 as 20 + 1 and 32 as 30 + 2. Use the four products to find 21 × 32.

Answer: 672

600 + 40 + 30 + 2 = 672.

When a becomes a + m and b becomes b + n, the increase is an + bm + _. What is the missing part?

• xy + 5x + 4y. You have x × y, x × 5 and 4 × y. The fourth product, 4 × 5 = 20, is missing.
• xy + 5x + 4y + 20 — correct. Yes! x × y, x × 5, 4 × y and 4 × 5: every term meets every term.
• xy + 20. That multiplies only the first terms and the last terms. x × 5 and 4 × y are missing.
• ab. ab is the old product. The increase is only what was added.
• m + n. The parts are products of terms, not sums.
• mn — correct. Yes! The small corner cell m × n.