Question

Please help expand the expression using pascal's triangle

(1-2x)^4

Answer 1

(1 - 2x)⁴
(1 - 2x)(1 - 2x)(1 - 2x)(1 - 2x)
[1(1 - 2x) - 2x(1 - 2x)][1(1 - 2x) - 2x(1 - 2x)]
[1(1) - 1(2x) - 2x(1) - 2x(-2x)][1(1) - 1(2x) - 2x(1) - 2x(-2x)]
(1 - 2x - 2x + 4x²)(1 - 2x - 2x + 4x²)
(1 - 4x + 4x²)(1 - 4x + 4x²)
1(1 - 4x + 4x²) - 4x(1 - 4x + 4x²) + 4x²(1 - 4x + 4x²)
1(1) - 1(4x) + 1(4x²) - 4x(1) - 4x(-4x) - 4x(4x²) + 4x²(1) - 4x²(4x) + 4x²(4x²)
1 - 4x + 4x² - 4x + 16x² - 16x³ + 4x² - 16x³ + 16x⁴
1 - 4x - 4x + 4x² + 16x² + 4x² - 16x³ - 16x³ + 16x⁴
1 - 8x + 24x² - 32x³ + 16x⁴

Answer 2

Expanding (1-2x)⁴ would get you something like

? (-2x)⁰1⁴ + ? (-2x)¹ 1³ + ? (-2x)² 1² + ? (-2x)³ 1¹ + ? (-2x)⁴ 1⁰

? - ? 2x + ? 4x² - ? 8x³ + ? 16x⁴

Fourth row in pascal triangle would be

1, 4, 6, 4, 1

So you insert these numbers in " ? "

1 - 4·2x + 6·4x² - 4·8x³ + 1·16x⁴

Simplify a little more;

(1-2x)⁴ = 1 - 8x + 24x² - 32x³ + 16x⁴