Multiplying Brackets ഗുണനസമവാക്യങ്ങൾ
When every side of a rectangle grows by x, the new area is a product like (x + 3)(x + 2). To turn such a product into a polynomial we “open the brackets”. This page shows how, with area pictures that make every term visible – exactly the tool the textbook uses on pages 178–180.
- Letters for numbers and like terms (page 1); x² and x³ (page 2).
- Area of a rectangle = length × width.
1. One number times a sum
Take 3 rows of (x + 4) chairs: there are 3 rows of x chairs and 3 rows of 4 chairs, so 3(x + 4) = 3x + 12. Multiplying a sum means multiplying each part and adding. This is the distributive law.
2. A sum times a sum: (x + a)(x + b)
Now both sides are sums. A rectangle with sides x + 3 and x + 2 splits into four pieces: x², 2x, 3x and 6. Add them and collect like terms.
(x + a)(x + b) = x² + (a + b)x + ab
The middle coefficient is the sum of a and b; the last number is their product. (x + 3)(x + 2): 3 + 2 = 5 and 3 × 2 = 6.
The check line puts a number in for x on both sides. They always agree – that is what “identity” means.
3. Three terms times two terms
For the volume of a block whose edges grow by x, the textbook multiplies (x + 1) by (x² + 5x + 6). Rule: multiply each term of one bracket by each term of the other – 2 × 3 = 6 products – then collect like terms. A grid keeps the work tidy.
Put x = 1: left side (1 + 1)(1 + 5 + 6) = 2 × 12 = 24; right side 1 + 6 + 11 + 6 = 24 ✔. Testing with a small number is a quick way to catch mistakes.
4. Minus signs and (a + b)(a − b)
Negative terms follow the sign rules: (x − 2)(x + 5) = x² + 5x − 2x − 10 = x² + 3x − 10. A special case is worth remembering:
13 × 7 = (10 + 3)(10 − 3) = 100 − 9 = 91. And 52 × 48 = 2500 − 4 = 2496. In the chapter this identity makes a square root vanish: (x + √x)(x − √x) = x² − (√x)² = x² − x.
5. Check yourself
- Multiply each term of one bracket by each term of the other, then collect like terms.
- (x + a)(x + b) = x² + (a + b)x + ab; (a + b)(a − b) = a² − b².
- Write the answer from the highest power down: x³ + 6x² + 11x + 6.
- Check by putting a small number such as x = 1 into both sides.