Polynomials
Senior 2 · Mathematics
Polynomials
Lesson 4 of 5
22 min
Can you work out 992 in your head? Write it as (100 − 1)2 = 10,000 − 200 + 1 = 9,801. That shortcut is an identity, a rule that is true for every value of the letters.
Identities save time in calculation and, more importantly, they run backwards: reading an identity from right to left is how you factorise, which is the subject of the next lesson.
An expression of the form ax2 + bx + c with a ≠ 0 is a quadratic expression. The identities below all produce quadratics.
1. Square of a sum. (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2.
The diagram shows why: a square of side a + b is made of a square a2, two rectangles ab and a square b2.
2. Square of a difference. (a − b)2 = a2 − 2ab + b2.
3. Difference of two squares. (a + b)(a − b) = a2 − ab + ab − b2 = a2 − b2.
A fourth, more general one: (x + p)(x + q) = x2 + (p + q)x + pq. The middle coefficient is the sum of p and q; the constant is their product.
1. (3x − 2)2: here a = 3x and b = 2. a2 = 9x2, 2ab = 12x, b2 = 4. Answer: 9x2 − 12x + 4.
2. (2x + 5)(2x − 5) = (2x)2 − 52 = 4x2 − 25.
3. (x + 3)(x + 4) = x2 + 7x + 12, since 3 + 4 = 7 and 3 × 4 = 12.
Mental arithmetic. 1022 = (100 + 2)2 = 10,000 + 400 + 4 = 10,404. And 47 × 53 = (50 − 3)(50 + 3) = 2,500 − 9 = 2,491.
Area. A square plot in Nyagatare has sides of 102 m. By the first identity its area is 10,404 m2, found without long multiplication.
Finding unknown combinations. If a + b = 7 and ab = 10, then a2 + b2 = (a + b)2 − 2ab = 49 − 20 = 29.
Tip
Exam technique. The most common error in all of algebra is writing (a + b)2 = a2 + b2. Test it with numbers: (3 + 4)2 = 49, but 32 + 42 = 25. The missing 24 is exactly 2ab, the two rectangles in the diagram.
Expand (x + 6)2.
Solution: x2 + 12x + 36.
Expand (5 − 2y)2.
Solution: 25 − 20y + 4y2.
Use an identity to work out 1012 and 992.
Solution: 1012 = 10,000 + 200 + 1 = 10,201. 992 = 10,000 − 200 + 1 = 9,801.
Is (x − 1)(x + 1) = x2 − 1 an identity or an equation?
Solution: An identity: it is the difference of two squares and holds for every x.
If a − b = 3 and ab = 4, find a2 + b2.
Solution: (a − b)2 = a2 − 2ab + b2, so a2 + b2 = 9 + 8 = 17.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
