Algebra - Factorising By Grouping

Solutions

Question 1

a) $\begin{align}(a+b)(x+y)&=a(x+y)+b(x+y)\\&=ax+ay+bx+by\end{align}$

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b) $\begin{align}(a+2)(b+3)&=a(b+3)+2(b+3)\\&=ab+3a+2b+6\end{align}$

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c) $\begin{align}(x+5)(x+8)&=x(x+8)+5(x+8)\\&=x^2+8x+5x+40\\&=x^2+13x+40\end{align}$

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d) $\begin{align}(x^2+x)(x+10)&=x^2(x+10)+x(x+10)\\&=x^3+10x^2+x^2+10x\\&=x^3+11x^2+10x\end{align}$

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e) $\begin{align}(a-x)(b+y)&=a(b+y)-x(b+y)\\&=ab+ay-bx-xy\end{align}$

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f) $\begin{align}(x-6)(y-9)&=x(y-9)-6(y-9)\\&=xy-9x-6y+54\end{align}$

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Question 2

a) $\begin{align}px+py+qx+qy&=p(x+y)+q(x+y)\\&=(p+q)(x+y)\end{align}$

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b) $\begin{align}sp+sq+tp+tq&=s(p+q)+t(p+q)\\&=(s+t)(p+q)\end{align}$

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c) $\begin{align}ab+5a+3b+15&=a(b+5)+3(b+5)\\&=(a+3)(b+5)\end{align}$

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d) $\begin{align}xy+10x+6y+60&=x(y+10)+6(y+10)\\&=(x+6)(y+10)\end{align}$

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e) $\begin{align}x^2+8x+3x+24&=x(x+8)+3(x+8)\\&=(x+3)(x+8)\end{align}$

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f) $\begin{align}y^2+7y+5y+35&=y(y+7)+5(y+7)\\&=(y+5)(y+7)\end{align}$

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Question 3

a) $\begin{align}ax+ay-bx-by&=a(x+y)-b(x+y)\\&=(a-b)(x+y)\end{align}$

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b) $\begin{align}hm+hn-5m-5n&=h(m+n)-5(m+n)\\&=(h-5)(m+n)\end{align}$

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c) $\begin{align}cx-2c+dx-2d&=c(x-2)+d(x-2)\\&=(c+d)(x-2)\end{align}$

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d) $\begin{align}py-pz+2y-2z&=p(y-z)+2(y-z)\\&=(p+2)(y-z)\end{align}$

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e) $\begin{align}x^2+7x-2x-14&=x(x+7)-2(x+7)\\&=(x-2)(x+7)\end{align}$

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f) $\begin{align}x^2-x+7x-7&=x(x-1)+7(x-1)\\&=(x+7)(x-1)\end{align}$

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Question 4

a) $\begin{align}ac-ad-bc+bd&=a(c-d)-b(c-d)\\&=(a-b)(c-d)\end{align}$

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b) $\begin{align}hx-2x-hy+2y&=x(h-2)-y(h-2)\\&=(x-y)(h-2)\end{align}$

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c) $\begin{align}xy-x^2-6y+6x&=x(y-x)-6(y-x)\\&=(x-6)(y-x)\end{align}$

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d) $\begin{align}5v^2-hv^2-5w+hw&=v^2(5-h)-w(5-h)\\&=(v^2-w)(5-h)\end{align}$

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e) $\begin{align}x^2-4x-6x+24&=x(x-4)-6(x-4)\\&=(x-6)(x-4)\end{align}$

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f) $\begin{align}z^2-8z-z+8&=z(z-8)-1(z-8)\\&=(z-1)(z-8)\end{align}$

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Question 5

a) $\begin{align}xy-2x+5y-10&=x(y-2)+5(y-2)\\&=(x+5)(y-2)\end{align}$

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b) $\begin{align}px-py-qx+qy&=p(x-y)-q(x-y)\\&=(p-q)(x-y)\end{align}$

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c) $\begin{align}3x^2+15x+4x+20&=3x(x+5)+4(x+5)\\&=(3x+4)(x+5)\end{align}$

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d) $\begin{align}2x^2+12x-5x-30&=2x(x+6)-5(x+6)\\&=(2x-5)(x+6)\end{align}$

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e) $\begin{align}4ap-4bp+5a-5b&=4p(a-b)+5(a-b)\\&=(4p+5)(a-b)\end{align}$

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f) $\begin{align}5t^2-5t-6t+6&=5t(t-1)-6(t-1)\\&=(5t-6)(t-1)\end{align}$

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g) $\begin{align}6xy-3x-8y+4&=3x(2y-1)-4(2y-1)\\&=(3x-4)(2y-1)\end{align}$

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h) $\begin{align}40z^2-8z-5z+1&=8z(5z-1)-1(5z-1)\\&=(8z-1)(5z-1)\end{align}$

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Question 6

a) $\begin{align}20pq+24p+15q+18&=4p(5q+6)+3(5q+6)\\&=(4p+3)(5q+6)\end{align}$

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b) $\begin{align}9t^3+27t^2-7t-21&=9t^2(t+3)-7(t+3)\\&=(9t^2-7)(t+3)\end{align}$

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c) $\begin{align}4x+12-x^3-3x^2&=4(x+3)-x^2(x+3)\\&=(4-x^2)(x+3)\end{align}$

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d) $\begin{align}2x^3-9x^2-8x+36&=x^2(2x-9)-4(2x-9)\\&=(x^2-4)(2x-9)\end{align}$

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e) $\begin{align}3x^4-15x^2+x^2-5&=3x^2(x^2-5)+1(x^2-5)\\&=(3x^2+1)(x^2-5)\end{align}$

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f) $\begin{align}20z^6-8z^3-5z^3+2&=4z^3(5z^3-2)-1(5z^3-2)\\&=(4z^3-1)(5z^3-2)\end{align}$

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Question 7

a) $\begin{align}2xy+10x+6y+30&=2(xy+5x+3y+15)\\&=2(x(y+5)+3(y+5))\\&=2(x+3)(y+5)\end{align}$

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b) $\begin{align}3x^2-27x+6x-54&=3(x^2-9x+2x-18)\\&=3(x(x-9)+2(x-9))\\&=3(x+2)(x-9)\end{align}$

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c) $\begin{align}2x^3+8x^2+5x^2+20x&=x(2x^2+8x+5x+20)\\&=x(2x(x+4)+5(x+4))\\&=x(2x+5)(x+4)\end{align}$

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d) $\begin{align}12xyz-12xy-9xz+9x&=3x(4yz-4y-3z+3)\\&=3x(4y(z-1)-3(z-1))\\&=3x(4y-3)(z-1)\end{align}$

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Question 8

a) $\begin{align}x^3+2x^2-25x+2&=x^2(x+2)-25(x+2)\\&=(x^2-25)(x+2)\\&=(x-5)(x+5)(x+2)\end{align}$

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b) $\begin{align}x^3+5x^2-8x-20&=x^2(2x+5)-4(2x+5)\\&=(x^2-4)(2x+5)\\&=(x-2)(x+2)(2x+5)\end{align}$

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c) $\begin{align}9x^3+54x^2-x-6&=9x^2(x+6)-1(x+6)\\&=(9x^2-1)(x+6)\\&=(3x-1)(3x+1)(x+6)\end{align}$

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d) $\begin{align}32x^3-80x^2-18x+45&=16x^2(2x-5)-9(2x-5)\\&=(16x^2-9)(2x-5)\\&=(4x-3)(4x+3)(2x-5)\end{align}$

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Question 9

a) $\begin{align}ax+bx+ay+by+az+bz&=x(a+b)+y(a+b)+z(a+b)\\&=(x+y+z)(a+b)\end{align}$

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b) $\begin{align}x^3-4x^2+x^2-4x+3x-12&=x^2(x-4)+x(x-4)+3(x-4)\\&=(x^2+x+3)(x-4)\end{align}$

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c) $\begin{align}6ax-3bx-2ay+by+8az-4bz&=3x(2a-b)-y(2a-b)+4z(2a-b)\\&=(3x-y+4z)(2a-b)\end{align}$

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d) $\begin{align}8x^3-16x^2-20x^2+40x-12x+24&=4(2x^3-4x^2-5x^2+10x-3x+6)\\&=4(2x^2(x-2)-5x(x-2)-3(x-2))\\&=4(2x^2-5x-3)(x-2)\end{align}$

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Question 10

a) $\begin{align}ax+bx+cx+ay+by+cy&=x(a+b+c)+y(a+b+c)\\&=(x+y)(a+b+c)\end{align}$

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b) $\begin{align}x^3+x^2y+2x^2-xy^2-y^2-2y^2&=x^2(x+y+2)-y^2(x+y+2)\\&=(x^2-y^2)(x+y+2)\\&=(x-y)(x+y)(x+y+2)\end{align}$

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