4.1.13 - the binomial theorem ii · the binomial theorem the binomial theorem tells us that: (x...

4.1.13 - The Binomial Theorem II

4.1 - Algebra - Expressions

Leaving Certificate Mathematics

Higher Level ONLY

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 1 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)

xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xn

y0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0

+

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1

+

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2

+ . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r

+ . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1

+

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

The Binomial Theorem

The Binomial Theorem tells us that:

(x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + . . .

. . . +

(n

r

)xn−ry r + . . .+

(n

n − 1

)x1yn−1 +

(n

n

)x0yn

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 2 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0

+

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1

+

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2

+ . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3

+

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1)

+ 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3)

+ 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9)

+ 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27)

+ . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4

− 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3

+ 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2

− 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x

+ 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3

Example 1

Q. Expand (2x − 3)4.

Answer:

(2x − 3)4 =

(4

0

)(2x)4(−3)0 +

(4

1

)(2x)3(−3)1 +

(4

2

)(2x)2(−3)2 + . . .

. . . +

(4

3

)(2x)1(−3)3 +

(4

4

)(2x)0(−3)4

= 1(16x4)(1) + 4(8x3)(−3) + 6(4x2)(9) + 4(2x)(−27) + . . .

. . . +1(1)(81)

= 16x4 − 96x3 + 216x2 − 216x + 81

4.1 - Algebra - Expressions 4.1.13 - The Binomial Theorem II Higher Level ONLY 3 / 3