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