Please show work and answer please!!

Question

Expanding each integer in the expression 21 × 12, the result is [(2 × 101) + (1 × 100)] × [(1 × 101) + (2 × 100)]. Replace the 10s with x’s and then multiply this expression using FOIL. Compare it with the normal integer multiplication method used above: How are they similar? Show your work.

Answer 1

Assuming you mean 101 = 10^1 and 100 = 10^0 (we use "^" to denote power)...

Let's first see the normal integer multiplication method

21 * 12 = [(2 × 10^1) + (1 × 10^0)] × [(1 × 10^1) + (2 × 10^0)] = 252

So 21 * 12 = 252

Now let's do this other method.

You already did the first step of [(2 × 10^1) + (1 × 10^0)] × [(1 × 10^1) + (2 × 10^0)]

Now let's substitute the 10s with the xs,

[(2 × x^1) + (1 × x^0)] × [(1 × x^1) + (2 × x^0)]

Since x^0 = 1 and x^1 = x,

(2x + 1) (x + 2)

Now let's do the FOIL method,

(2x + 1) (x + 2) = 2x*x + 2x*2 + 1*x + 1*2

= 2x^2 + 5x + 2

And now let's subsitute x=10 again,

2(10)^2 + 5(10) + 2 = 252

See anything similar?

Answer 2

21 x 12 = 21x2 + 21x10 = 42 +210 = 252 is more normal long division

21

x12

42

21

252

or 2x1 + 2x20 + 10x1 + 10x20 = 2+40+10+200 = 252

replace 10's by x's to get 2 +4x + x + x(2x) = 2x^2 +5x +2

reverse the substition and replace x by 10 gives 2(10)^2 +5(10) +2 = 200+50+2 =252

your alternative method is 21 = 2x10 + 1 and 12 = 1x10 + 2, so 21x12 = (2(10)+1)(10+2) replace 10 by x = (2x+1)(x+2) = 2x^2 +x + 4x +2 = 2x^2 +5x + 2:

or 21 x 12 = [2x10^1 + 1 x10^0] x [1 x 10^1 + 2 x10^0] = 21x 12 = 252

replace 10 by x,

[2(x) + 1(1)][(1)(x) + 2(1)] = (2x+1)(x+2) = 2x^2 +5x +2

replace x by 10, reversing the previous replacement of 10 by x, to get 2(10)^2 +5(10) +2 = 200+50+2 = 252

two alternative ways, although not that different actually, to get the same answer