Special Products and Expansions

Mathematical open statements are statements like x +1 = 3 which may be true or false depending upon the value of x.
Identities are statements like (x -1)² = x² -2x +1, which are true for all values of x.
Expansions are 2nd or higher powers of algebraic expressions.
You learnt the following special products:
(x +a)(x +b) = x² +(a +b)x +ab
(x -a)(x -b) = x² -(a +b)x +ab
(x +a)² = x² +2ax +a²
(x -a)² = x² -2ax +a²
(x +a)(x -a) = x² -a²
= a²+1/a² +2
= a² +1/a²-2.

Exercise

Simplify the following:
(i)
(ii)
(iii) (2 +x)(2 -x)
(iv) (2x +3)(2x +4)
(v) (3x -1)(3x -2)
(vi) (3s -4t)²
(vii) (a² +b²)²
(viii)
(ix) (x +y +z)(x +y -z)
(x)
(xi) (2 +x)²
(xii) (2 +3)²
If a -1/a = 3, find
(i) a² +1/a²
(ii) a⁴ +1/a⁴
If a +1/a = 3, find
(i) a² +1/a²
(ii) a⁴ +1/a⁴
Simplify the following:
(i) (a +b)² +(a -b)²
(ii) (a +b)² -(a -b)²
(iii)
(iv)
Use (a +b)² = a² +2ab +b² to evaluate
(i) (101)²
(ii) (10.2)²
Use (a -b)² = a² -2ab +b² to evaluate
(i) (997)²
(ii) (9.8)²
Evaluate 1002×998 by using a special product.
Fill in the blanks in the following:
(i) (2x -...)² = 4x² -... +9y²
(ii) (a +...)² =... +... +4b²
(iii) (3s +2t)(3s -2t) =... -...
Simplify .
Simplify (x -2)(x +2)(x² +4)(x⁴ +16).
Simplify
If x -1/x = 5, find the value of x² +1/x²

Answers

1. (i) 4a² -b²/4 (ii) x² -1/4   (iii) 4 -x²
    (iv) 4x² +14x +12             (v) 9x² -9x +2
    (vi) 9s² -24st +16t²          (vii) a⁴ +2a²b² +b⁴
    (viii) 9a⁴ -2 +1/9a⁴               (ix) x² +2xy +y² -z² (x) x²/2 +8/y² +4x/y
    (xi) 2 +2

2x +x² (xii) 5 +2

6.
2. (i) 11 (ii) 119 3. (i)7 (ii) 47
4. (i) 2 (a² +b²) (ii) 4ab (iii)

(iv) 4                          5. (i) 10201      (ii) 104.04.
6. (i) 994009               (ii) 96.04          7. 999996
8. (i) 3y, 12xy               (ii) 2b,a², 4ab (iii) 9s² , 4t²
9. 1/2 x² -2                10. x⁸ -256
11. x² +1/x²+3 -2x -2/x   12. 7