Figure It Out: Grid & Expansions

1. Multiplication Grid (Page 142)

The grid represents a multiplication table. We are given a $3 \times 3$ frame centered at $pq$. If the center is $pq$ (where $p$ is row number, $q$ is column number), the surrounding cells are:

2. Expansions (Page 143)

(i) $(3+u)(v-3)$ $= 3(v-3) + u(v-3)$ $= 3v - 9 + uv - 3u$

(ii) $\frac{2}{3}(15 + 6a)$ $= \frac{2}{3} \times 15 + \frac{2}{3} \times 6a$ $= 2 \times 5 + 2 \times 2a$ $= 10 + 4a$

(iii) $(10a + b)(10c + d)$ $= 100ac + 10ad + 10bc + bd$

(iv) $(3-x)(x-6)$ $= 3x - 18 - x^2 + 6x$ $= -x^2 + 9x - 18$

3. Product Unchanged

Find $a, b$ such that $(a+2)(b-4) = ab$. $ab - 4a + 2b - 8 = ab$ $2b - 4a = 8 \implies b - 2a = 4 \implies b = 2a + 4$ Examples:

1. $a=1, b=6 \rightarrow 1 \times 6 = 6; (3)(2) = 6$.
2. $a=2, b=8 \rightarrow 2 \times 8 = 16; (4)(4) = 16$.
3. $a=3, b=10 \rightarrow 3 \times 10 = 30; (5)(6) = 30$.

5. Expansion Patterns

(i) $(a-b)(a+b) = a^2 - b^2$ (ii) $(a-b)(a^2+ab+b^2) = a^3 - b^3$ (iii) $(a-b)(a^3+a^2b+ab^2+b^3) = a^4 - b^4$

Next Identity: $(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4) = a^5 - b^5$.