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Pascals Triangle Worksheet

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Mathematics
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Worksheet
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1. (x + 2)^6
The seventh row of Pascal’s triangle would be 1, 6, 15, 20, 15, 6, and 1
1x^6 2^0 + 6x^5 2^1 + 15x^4 2^2 + 20x^3 2^3 + 15x^2 2^4 + 6x^1 2^5 + 1x^0 + 2^6
x^6 + 6*2x^5 + 15*4x^4 + 20*8x^3 + 15*16x^2 + 6*32 + 64
x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192 + 64
2. (x − 4)^4
The fifth row of Pascal’s triangle would be 1, 4, 6, 4, and 1
1x^4 -4^0 + 4x^3 -4^1 + 6x^2 -4^2 + 4x^1 -4^3 + 1x^0 -4^4
x^4 + 4*-4x^3 + 6*16x^2 + 4*-64x + 256
x^4 - 16x^3 + 96x^2 – 256x + 256
3. (2x + 3)^5
The sixth row of Pascal’s triangle would be 1, 5, 10, 10, 5, 1
1(2x)^5 3^0 + 5(2x)^4 3^1 + 10(2x)^3 3^2 + 10(2x)^2 3^3 + 5(2x)^1 3^4 + 1(2x)^0 3^5
2x^5 + 80*3x^4 + 80*9x^3 + 40*27x^2 + 10*81x + 243
2x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243
32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243
4. (2x − 3y)^4
The fifth row of Pascal’s triangle would be 1, 4, 6, 4, and 1
1(2x)^4 -3y^0 + 4(2x)^3 -3y^1 + 6(2x)^2 -3y^2 + 4(2x)^1 -3y^3 + 1(2x)^0 -3y^4
2x^4 + 32x^3 -3y + 24x^2 + 9y^2 + 8x -27y^3 + 81y^4
16x^4 - 96x^3y + 216x^2y^2 – 216xy^3 + 81y^4
5. In the expansion of (3a + 4b)^8, which of the following are possible variable terms? Explain your
reasoning. a^2b^3; a^5b^3; ab^8; b^8; a^4b^4; a^8; ab^7; a^6b^5
If we use the binomial theorem that says (a + b)
1
= a + b (a + b)
2
= (a + b)(a + b) = a
2
+ 2ab + b
2
(a
+ b)
3
= (a + b)(a
2
+ 2ab + b
2
) = a
3
+ 3a
2
b + 3ab
2
+ b
3
and so on we will end up with
a^8,a^5b^3,a^4b^4,ab^7, and b^8
Essential Question:
How does the Binomial Theorem’s use Pascal’s triangle to expand binomials raised to positive integer
powers?
The Pascal's Triangle gives us the coefficients for the Binomial Theorem such as (a + b)^4 = 1a^4b^0 +
4a^3b^1 + 6a^2b^2 + 4a^^1b3 + 1a^0b^4 = a^4 + 4a^3b^1 + 6a^2b^2 + 4a^1b^3 + b^4

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1. (x + 2)^6 The seventh row of Pascal’s triangle would be 1, 6, 15, 20, 15, 6, and 1 1x^6 2^0 + 6x^5 2^1 + 15x^4 2^2 + 20x^3 2^3 + 15x^2 2^4 + 6x^1 2^5 + 1x^0 + 2^6 x^6 + 6*2x^5 + 15*4x^4 + 20*8x^3 + 15*16x^2 + 6*32 + 64 x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192 + 64 2. (x − 4)^4 The fifth ro ...
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