We Distribute, Yet Things Multiply

Expand the following products. (i) $(3 + u)(v - 3)$ (ii) $\frac{2}{3}(15 + 6a)$ (iii) $(10a + b)(10c + d)$ (iv) $(3 - x)(x - 6)$ (v) $(-5a + b)(c + d)$ (vi) $(5 + z)(y + 9)$

Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

Expand (i) $(a + ab - 3b^2)$ $(4 + b)$, and (ii) $(4y + 7)$ $(y + 11z - 3)$.

Expand (i) $(a - b)(a + b)$, (ii) $(a - b)(a^2 + ab + b^2)$ and (iii) $(a - b)(a^3 + a^2b + ab^2 + b^3)$. Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

Which is greater: $(a - b)^2$ or $(b - a)^2$? Justify your answer.

Express 100 as the difference of two squares.

Find $406^2, 72^2, 145^2, 1097^2$, and $124^2$ using the identities you have learnt so far.

Do Patterns 1 and 2 hold only for counting numbers? Do they hold for negative integers as well? What about fractions? Justify your answer.

$-3p(-5p + 2q) = -3p + 5p - 2q = p - 2q$

$2(x - 1) + 3(x + 4) = 2x - 1 + 3x + 4 = 5x + 3$

$y + 2(y + 2) = (y + 2)^2 = y^2 + 4y + 4$

$(5m + 6n)^2 = 25m^2 + 36n^2$

$(-q + 2)^2 = q^2 - 4q + 4$

$3a(2b \times 3c) = 6ab \times 9ac = 54a2bc$

$\frac{1}{2}(10s - 6) + 3 = 5s - 3 + 3 = 5s$

$5w^2 + 6w = 11w^2$

$2a^3 + 3a^3 + 6a^2b + 6ab^2 = 5a^3 + 12a^2b^2$

$(x + 2)(x + 5) = (x + 2)x + (x + 2)5 = x^2 + 2x + 5x + 10 = x^2 + 7x + 10$

$(a + 2)(b + 4) = ab + 8$

$ab^2 + a^2b + a^2b^2 = ab(a + b + ab)$