Mathematics Worksheet/Algebra/Pascal's Triangle

Study the Pascal's Triangle below.

                                         1
                                       1   1
                                     1   2   1
                                   1   3   3   1
                                 1   4   6   4   1
                               1   5  10  10   5   1

This table showing the $(x+y{)}^n$, exponential results and coefficients.

$(x+y{)}^n$	Exponential results	Coefficients
$(x+y)$	$x+y$	1 1
$(x+y{)}^2$	$x^2+2xy+y^2$	1 2 1
$(x+y{)}^3$	$x^3+3x^{2}y+3x{y}^2+y^3$	1 3 3 1
$(x+y{)}^4$	$x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+y^4$	1 4 6 4 1
$(x+y{)}^5$	$x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^3+5x{y}^4+y^5$	1 5 10 10 5 1

Now do the questions carefully using Pascal's triangle; Use ^ to write exponentiation. <quiz display=simple points="1/1">

{

<charinsert> α Δ</charinsert>

}

{ |type="{}"} $(y+3{)}^2=${ y^2 + 6y + 9 (i)|y^2+6y+9 (i) _144 }

{ |type="{}"} $(y-5{)}^2=${ y^2 - 10y + 25 (i)|y^2-10y+25 (i) _144 }

{ |type="{}"} $(\mathrm{\Delta }+4{)}^2=${ Δ^2 + 8Δ + 16 (i)|Δ^2+8Δ+16 (i) _144 }

{ |type="{}"} $(3x+5{)}^3=${ 27x^3 + 135x^2 + 225x + 125 (i)|27x^3+135x^2+225x+125 (i) _144 }

{ |type="{}"} $(4x-2y{)}^3=${ 64x^3 - 96x^2y + 48xy^2 - 8y^3 (i)|64x^3-96x^2y+48xy^2-8y^3 (i) _144 }

{ |type="{}"} $(\alpha +y{)}^2=${ α^2 + 2αy + y^2 (i)|α^2+2αy+y^2 (i) _144 }

{ |type="{}"} $(5x^2+3x^3{)}^4=${ 625x^8 + 1,500x^9 + 1,350x^10 + 540x^11 + 81x^12 (i)|625x^8+1,500x^9+1,350x^10+540x^11+81x^12 (i)|81x^12 + 540x^11 + 1,350x^10 + 1,500x^9 + 625x^8 (i)|81x^12+540x^11+1,350x^10+1,500x^9+625x^8 (i)|625x^8 + 1500x^9 + 1350x^10 + 540x^11 + 81x^12 (i)|625x^8+1500x^9+1350x^10+540x^11+81x^12 (i)|81x^12 + 540x^11 + 1350x^10 + 1500x^9 + 625x^8 (i)|81x^12+540x^11+1350x^10+1500x^9+625x^8 (i) _144 }

{ |type="{}"} $(3x+2y-5z{)}^2=${ 9x^2 + 12xy - 30xz + 4y^2 - 20yz + 25z^2 (i)|9x^2+12xy-30xz+4y^2-20yz+25z^2 (i) _144 }

{ |type="{}"} $(5x^3-3y^5{)}^4=${ 625x^12 - 1,500x^9y^5 + 1,350x^6y^10 - 540x^3y^15 + 81y^20 (i)|625x^12-1,500x^9y^5+1,350x^6y^10-540x^3y^15+81y^20 (i)|625x^12 - 1500x^9y^5 + 1350x^6y^10 - 540x^3y^15 + 81y^20 (i)|625x^12-1500x^9y^5+1350x^6y^10-540x^3y^15+81y^20 (i)|81y^20 - 540x^3y^15 + 1,350x^6y^10 - 1,500x^9y^5 + 625x^12 (i)|81y^20-540x^3y^15+1,350x^6y^10-1,500x^9y^5+625x^12 (i)|81y^20 - 540x^3y^15 + 1350x^6y^10 - 1500x^9y^5 + 625x^12 (i)|81y^20-540x^3y^15+1350x^6y^10-1500x^9y^5+625x^12 (i) _144 }

{ |type="{}"} $(18x+3y^4{)}^6=${ 34,012,224x^6 + 34,012,224x^5y^4 + 14,171,760x^4y^8 + 3,149,280x^3y^12 + 393,660x^2y^16 + 26,244xy^20 + 729y^24 (i)|34,012,224x^6+34,012,224x^5y^4+14,171,760x^4y^8+3,149,280x^3y^12+393,660x^2y^16+26,244xy^20+729y^24 (i)|34012224x^6 + 34012224x^5y^4 + 14171760x^4y^8 + 3149280x^3y^12 + 393660x^2y^16 + 26244xy^20 + 729y^24 (i)|34012224x^6+34012224x^5y^4+14171760x^4y^8+3149280x^3y^12+393660x^2y^16+26244xy^20+729y^24 (i)|729y^24 + 26,244xy^20 + 393,660x^2y^16 + 3,149,280x^3y^12 + 14,171,760x^4y^8 + 34,012,224x^5y^4 + 34,012,224x^6 (i)|729y^24+26,244xy^20+393,660x^2y^16+3,149,280x^3y^12+14,171,760x^4y^8+34,012,224x^5y^4+34,012,224x^6 (i)|729y^24 + 26244xy^20 + 393660x^2y^16 + 3149280x^3y^12 + 14171760x^4y^8 + 34012224x^5y^4 + 34012224x^6 (i)|729y^24+26244xy^20+393660x^2y^16+3149280x^3y^12+14171760x^4y^8+34012224x^5y^4+34012224x^6 (i) _144 }

{ |type="{}"} $(a+9{)}^3=${ a^3 + 27a^2 + 243a + 729 (i)|a^3+27a^2+243a+729 (i) _144 }

{ |type="{}"} $(b-6{)}^5=${ b^5 - 30b^4 + 360b^3 - 2,160b^2 + 6,480b - 7,776 (i)|b^5-30b^4+360b^3-2,160b^2+6,480b-7,776 (i)|b^5 - 30b^4 + 360b^3 - 2160b^2 + 6480b - 7776 (i)|b^5-30b^4+360b^3-2160b^2+6480b-7776 (i) _144 }

{ |type="{}"} $(6m+3n{)}^3=${ 216m^3 + 324m^2n + 162mn^2 + 27n^3 (i)|216m^3+324m^2n+162mn^2+27n^3 (i) _144 }

{ |type="{}"} $(3x+9y{)}^2=${ 9x^2 + 54xy + 81y^2 (i)|9x^2+54xy+81y^2 (i) _144 }

{ |type="{}"} $(a+b{)}^5=${ a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5 (i)|a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5 (i) _144 }

{ |type="{}"} $(6+y{)}^6=${ 46,656 + 46,656y + 19,440y^2 + 4,320y^3 + 540y^4 + 36y^5 + y^6 (i)|46,656+46,656y+19,440y^2+4,320y^3+540y^4+36y^5+y^6 (i)|46656 + 46656y + 19440y^2 + 4320y^3 + 540y^4 + 36y^5 + y^6 (i)|46656+46656y+19440y^2+4320y^3+540y^4+36y^5+y^6 (i)|y^6 + 36y^5 + 540y^4 + 4,320y^3 + 19,440y^2 + 46,656y + 46,656 (i)|y^6+36y^5+540y^4+4,320y^3+19,440y^2+46,656y+46,656 (i)|y^6 + 36y^5 + 540y^4 + 4320y^3 + 19440y^2 + 46656y + 46656 (i)|y^6+36y^5+540y^4+4320y^3+19440y^2+46656y+46656 (i) _144 }

{ |type="{}"} $(x-8{)}^7=${ x^7 - 56x^6 + 1,344x^5 - 17,920x^4 + 143,360x^3 - 688,128x^2 + 1,835,008x - 2,097,152 (i)|x^7-56x^6+1,344x^5-17,920x^4+143,360x^3-688,128x^2+1,835,008x-2,097,152 (i)|x^7 - 56x^6 + 1344x^5 - 17920x^4 + 143360x^3 - 688128x^2 + 1835008x - 2097152 (i)|x^7-56x^6+1344x^5-17920x^4+143360x^3-688128x^2+1835008x-2097152 (i) _144 }

{ |type="{}"} $(y-6x{)}^8=${ y^8 - 48xy^7 + 1,008x^2y^6 - 12,096x^3y^5 + 90,720x^4y^4 - 435,456x^5y^3 + 1,306,368x^6y^2 - 2,239,488x^7y + 1,679,616x^8 (i)|y^8-48xy^7+1,008x^2y^6-12,096x^3y^5+90,720x^4y^4-435,456x^5y^3+1,306,368x^6y^2-2,239,488x^7y+1,679,616x^8 (i)|y^8 - 48xy^7 + 1008x^2y^6 - 12096x^3y^5 + 90720x^4y^4 - 435456x^5y^3 + 1306368x^6y^2 - 2239488x^7y + 1679616x^8 (i)|y^8-48xy^7+1008x^2y^6-12096x^3y^5+90720x^4y^4-435456x^5y^3+1306368x^6y^2-2239488x^7y+1679616x^8 (i)|1,679,616x^8 - 2,239,488x^7y + 1,306,368x^6y^2 - 435,456x^5y^3 + 90,720x^4y^4 - 12,096x^3y^5 + 1,008x^2y^6 - 48xy^7 + y^8 (i)|1,679,616x^8-2,239,488x^7y+1,306,368x^6y^2-435,456x^5y^3+90,720x^4y^4-12,096x^3y^5+1,008x^2y^6-48xy^7+y^8 (i)|1679616x^8 - 2239488x^7y + 1306368x^6y^2 - 435456x^5y^3 + 90720x^4y^4 - 12096x^3y^5 + 1008x^2y^6 - 48xy^7 + y^8 (i)|1679616x^8-2239488x^7y+1306368x^6y^2-435456x^5y^3+90720x^4y^4-12096x^3y^5+1008x^2y^6-48xy^7+y^8 (i) _144 }

{ |type="{}"} $(x+y+z{)}^2=${ x^2 + 2xy + 2xz + y^2 + 2yz + z^2 (i)|x^2+2xy+2xz+y^2+2yz+z^2 (i) _144 }

{ |type="{}"} $(a+b-c+d{)}^3=${ a^3 + 3a^2b - 3a^2c + 3a^2d + 3ab^2 - 6abc + 6abd + 3ac^2 - 6acd + 3ad^2 + b3 - 3b^2c + 3b^2d + 3bc^2 - 6bcd + 3bd^2 - c^3 + 3c^2d - 3cd^2 + d^3 (i)|a^3+3a^2b-3a^2c+3a^2d+3ab^2-6abc+6abd+3ac^2-6acd+3ad^2+b3-3b^2c+3b^2d+3bc^2-6bcd+3bd^2-c^3+3c^2d-3cd^2+d^3 (i) _144 } </quiz> Template:BookCat