Algebra - Factorising By Grouping
Question 1
Expand the following
a) $(a+b)(x+y)$
a
$ax+ay+bx+by$
Question ID: 20210010010
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$\begin{align}(a+b)(x+y)&=a(x+y)+b(x+y)\\&=ax+ay+bx+by\end{align}$
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b) $(a+2)(b+3)$
a
$ab+3a+2b+6$
Question ID: 20210010020
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$\begin{align}(a+2)(b+3)&=a(b+3)+2(b+3)\\&=ab+3a+2b+6\end{align}$
Question ID: 20210010020
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c) $(x+5)(x+8)$
a
$x^2+13x+40$
Question ID: 20210010030
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$\begin{align}(x+5)(x+8)&=x(x+8)+5(x+8)\\&=x^2+8x+5x+40\\&=x^2+13x+40\end{align}$
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d) $(x^2+x)(x+10)$
a
$x^3+11x^2+10x$
Question ID: 20210010040
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$\begin{align}(x^2+x)(x+10)&=x^2(x+10)+x(x+10)\\&=x^3+10x^2+x^2+10x\\&=x^3+11x^2+10x\end{align}$
Question ID: 20210010040
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e) $(a-x)(b+y)$
a
$ab+ay-bx-xy$
Question ID: 20210010050
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$\begin{align}(a-x)(b+y)&=a(b+y)-x(b+y)\\&=ab+ay-bx-xy\end{align}$
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f) $(x-6)(y-9)$
a
$xy-9x-6y+54$
Question ID: 20210010060
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$\begin{align}(x-6)(y-9)&=x(y-9)-6(y-9)\\&=xy-9x-6y+54\end{align}$
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Question 2
Factorise the following by grouping (in pairs)
a) $px+py+qx+qy$
a
$(p+q)(x+y)$
Question ID: 20210020010
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$\begin{align}px+py+qx+qy&=p(x+y)+q(x+y)\\&=(p+q)(x+y)\end{align}$
Question ID: 20210020010
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b) $sp+sq+tp+tq$
a
$(s+t)(p+q)$
Question ID: 20210020020
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$\begin{align}sp+sq+tp+tq&=s(p+q)+t(p+q)\\&=(s+t)(p+q)\end{align}$
Question ID: 20210020020
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c) $ab+5a+3b+15$
a
$(a+3)(b+5)$
Question ID: 20210020030
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$\begin{align}ab+5a+3b+15&=a(b+5)+3(b+5)\\&=(a+3)(b+5)\end{align}$
Question ID: 20210020030
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d) $xy+10x+6y+60$
a
$(x+6)(y+10)$
Question ID: 20210020040
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$\begin{align}xy+10x+6y+60&=x(y+10)+6(y+10)\\&=(x+6)(y+10)\end{align}$
Question ID: 20210020040
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e) $x^2+8x+3x+24$
a
$(x+3)(x+8)$
Question ID: 20210020050
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$\begin{align}x^2+8x+3x+24&=x(x+8)+3(x+8)\\&=(x+3)(x+8)\end{align}$
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f) $y^2+7y+5y+35$
a
$(y+5)(y+7)$
Question ID: 20210020060
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$\begin{align}y^2+7y+5y+35&=y(y+7)+5(y+7)\\&=(y+5)(y+7)\end{align}$
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Question 3
Factorise the following by grouping (in pairs)
a) $ax+ay-bx-by$
a
$(a-b)(x+y)$
Question ID: 20210030010
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$\begin{align}ax+ay-bx-by&=a(x+y)-b(x+y)\\&=(a-b)(x+y)\end{align}$
Question ID: 20210030010
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Question ID: 20210030010
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b) $hm+hn-5m-5n$
a
$(h-5)(m+n)$
Question ID: 20210030020
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$\begin{align}hm+hn-5m-5n&=h(m+n)-5(m+n)\\&=(h-5)(m+n)\end{align}$
Question ID: 20210030020
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c) $cx-2c+dx-2d$
a
$(c+d)(x-2)$
Question ID: 20210030030
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$\begin{align}cx-2c+dx-2d&=c(x-2)+d(x-2)\\&=(c+d)(x-2)\end{align}$
Question ID: 20210030030
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d) $py-pz+2y-2z$
a
$(p+2)(y-z)$
Question ID: 20210030040
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$\begin{align}py-pz+2y-2z&=p(y-z)+2(y-z)\\&=(p+2)(y-z)\end{align}$
Question ID: 20210030040
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e) $x^2+7x-2x-14$
a
$(x-2)(x+7)$
Question ID: 20210030050
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$\begin{align}x^2+7x-2x-14&=x(x+7)-2(x+7)\\&=(x-2)(x+7)\end{align}$
Question ID: 20210030050
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f) $x^2-x+7x-7$
a
$(x+7)(x-1)$
Question ID: 20210030060
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$\begin{align}x^2-x+7x-7&=x(x-1)+7(x-1)\\&=(x+7)(x-1)\end{align}$
Question ID: 20210030060
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Question 4
Factorise the following by grouping (in pairs)
a) $ac-ad-bc+bd$
a
$(a-b)(c-d)$
Question ID: 20210040010
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$\begin{align}ac-ad-bc+bd&=a(c-d)-b(c-d)\\&=(a-b)(c-d)\end{align}$
Question ID: 20210040010
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b) $hx-2x-hy+2y$
a
$(x-y)(h-2)$
Question ID: 20210040020
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$\begin{align}hx-2x-hy+2y&=x(h-2)-y(h-2)\\&=(x-y)(h-2)\end{align}$
Question ID: 20210040020
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c) $xy-x^2-6y+6x$
a
$(x-6)(y-x)$
Question ID: 20210040030
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$\begin{align}xy-x^2-6y+6x&=x(y-x)-6(y-x)\\&=(x-6)(y-x)\end{align}$
Question ID: 20210040030
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Question ID: 20210040030
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d) $5v^2-hv^2-5w+hw$
a
$(v^2-w)(5-h)$
Question ID: 20210040040
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$\begin{align}5v^2-hv^2-5w+hw&=v^2(5-h)-w(5-h)\\&=(v^2-w)(5-h)\end{align}$
Question ID: 20210040040
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e) $x^2-4x-6x+24$
a
$(x-6)(x-4)$
Question ID: 20210040050
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$\begin{align}x^2-4x-6x+24&=x(x-4)-6(x-4)\\&=(x-6)(x-4)\end{align}$
Question ID: 20210040050
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Question ID: 20210040050
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f) $z^2-8z-z+8$
a
$(z-1)(z-8)$
Question ID: 20210040060
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$\begin{align}z^2-8z-z+8&=z(z-8)-1(z-8)\\&=(z-1)(z-8)\end{align}$
Question ID: 20210040060
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Question 5
Factorise the following by grouping (in pairs)
a) $xy-2x+5y-10$
a
$(x+5)(y-2)$
Question ID: 20210050010
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$\begin{align}xy-2x+5y-10&=x(y-2)+5(y-2)\\&=(x+5)(y-2)\end{align}$
Question ID: 20210050010
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b) $px-py-qx+qy$
a
$(p-q)(x-y)$
Question ID: 20210050020
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$\begin{align}px-py-qx+qy&=p(x-y)-q(x-y)\\&=(p-q)(x-y)\end{align}$
Question ID: 20210050020
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c) $3x^2+15x+4x+20$
a
$(3x+4)(x+5)$
Question ID: 20210050030
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$\begin{align}3x^2+15x+4x+20&=3x(x+5)+4(x+5)\\&=(3x+4)(x+5)\end{align}$
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d) $2x^2+12x-5x-30$
a
$(2x-5)(x+6)$
Question ID: 20210050040
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$\begin{align}2x^2+12x-5x-30&=2x(x+6)-5(x+6)\\&=(2x-5)(x+6)\end{align}$
Question ID: 20210050040
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e) $4ap-4bp+5a-5b$
a
$(4p+5)(a-b)$
Question ID: 20210050050
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$\begin{align}4ap-4bp+5a-5b&=4p(a-b)+5(a-b)\\&=(4p+5)(a-b)\end{align}$
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f) $5t^2-5t-6t+6$
a
$(5t-6)(t-1)$
Question ID: 20210050060
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$\begin{align}5t^2-5t-6t+6&=5t(t-1)-6(t-1)\\&=(5t-6)(t-1)\end{align}$
Question ID: 20210050060
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g) $6xy-3x-8y+4$
a
$(3x-4)(2y-1)$
Question ID: 20210050070
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$\begin{align}6xy-3x-8y+4&=3x(2y-1)-4(2y-1)\\&=(3x-4)(2y-1)\end{align}$
Question ID: 20210050070
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h) $40z^2-8z-5z+1$
a
$(8z-1)(5z-1)$
Question ID: 20210050080
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$\begin{align}40z^2-8z-5z+1&=8z(5z-1)-1(5z-1)\\&=(8z-1)(5z-1)\end{align}$
Question ID: 20210050080
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Question 6
Factorise the following by grouping (in pairs)
a) $20pq+24p+15q+18$
a
$(4p+3)(5q+6)$
Question ID: 20210060010
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$\begin{align}20pq+24p+15q+18&=4p(5q+6)+3(5q+6)\\&=(4p+3)(5q+6)\end{align}$
Question ID: 20210060010
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Question ID: 20210060010
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b) $9t^3+27t^2-7t-21$
a
$(9t^2-7)(t+3)$
Question ID: 20210060020
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$\begin{align}9t^3+27t^2-7t-21&=9t^2(t+3)-7(t+3)\\&=(9t^2-7)(t+3)\end{align}$
Question ID: 20210060020
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c) $4x+12-x^3-3x^2$
a
$(4-x^2)(x+3)$
Question ID: 20210060030
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$\begin{align}4x+12-x^3-3x^2&=4(x+3)-x^2(x+3)\\&=(4-x^2)(x+3)\end{align}$
Question ID: 20210060030
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d) $2x^3-9x^2-8x+36$
a
$(x^2-4)(2x-9)$
Question ID: 20210060040
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$\begin{align}2x^3-9x^2-8x+36&=x^2(2x-9)-4(2x-9)\\&=(x^2-4)(2x-9)\end{align}$
Question ID: 20210060040
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e) $3x^4-15x^2+x^2-5$
a
$(3x^2+1)(x^2-5)$
Question ID: 20210060050
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$\begin{align}3x^4-15x^2+x^2-5&=3x^2(x^2-5)+1(x^2-5)\\&=(3x^2+1)(x^2-5)\end{align}$
Question ID: 20210060050
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Question ID: 20210060050
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f) $20z^6-8z^3-5z^3+2$
a
$(4z^3-1)(5z^3-2)$
Question ID: 20210060060
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$\begin{align}20z^6-8z^3-5z^3+2&=4z^3(5z^3-2)-1(5z^3-2)\\&=(4z^3-1)(5z^3-2)\end{align}$
Question ID: 20210060060
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Question 7
Factorise the following by grouping (in pairs)
a) $2xy+10x+6y+30$
a
$2(x+3)(y+5)$
Question ID: 20210070010
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$\begin{align}2xy+10x+6y+30&=2(xy+5x+3y+15)\\&=2(x(y+5)+3(y+5))\\&=2(x+3)(y+5)\end{align}$
Question ID: 20210070010
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b) $3x^2-27x+6x-54$
a
$3(x+2)(x-9)$
Question ID: 20210070020
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$\begin{align}3x^2-27x+6x-54&=3(x^2-9x+2x-18)\\&=3(x(x-9)+2(x-9))\\&=3(x+2)(x-9)\end{align}$
Question ID: 20210070020
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Question ID: 20210070020
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c) $2x^3+8x^2+5x^2+20x$
a
$x(2x+5)(x+4)$
Question ID: 20210070030
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$\begin{align}2x^3+8x^2+5x^2+20x&=x(2x^2+8x+5x+20)\\&=x(2x(x+4)+5(x+4))\\&=x(2x+5)(x+4)\end{align}$
Question ID: 20210070030
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Question ID: 20210070030
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d) $12xyz-12xy-9xz+9x$
a
$3x(4y-3)(z-1)$
Question ID: 20210070040
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$\begin{align}12xyz-12xy-9xz+9x&=3x(4yz-4y-3z+3)\\&=3x(4y(z-1)-3(z-1))\\&=3x(4y-3)(z-1)\end{align}$
Question ID: 20210070040
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Question 8
Factorise the following by grouping (in pairs)
a) $x^3+2x^2-25x+2$
a
$(x-5)(x+5)(x+2)$
Question ID: 20210080010
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$\begin{align}x^3+2x^2-25x+2&=x^2(x+2)-25(x+2)\\&=(x^2-25)(x+2)\\&=(x-5)(x+5)(x+2)\end{align}$
Question ID: 20210080010
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Question ID: 20210080010
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b) $x^3+5x^2-8x-20$
a
$(x-2)(x+2)(2x+5)$
Question ID: 20210080020
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$\begin{align}x^3+5x^2-8x-20&=x^2(2x+5)-4(2x+5)\\&=(x^2-4)(2x+5)\\&=(x-2)(x+2)(2x+5)\end{align}$
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c) $9x^3+54x^2-x-6$
a
$(3x-1)(3x+1)(x+6)$
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$\begin{align}9x^3+54x^2-x-6&=9x^2(x+6)-1(x+6)\\&=(9x^2-1)(x+6)\\&=(3x-1)(3x+1)(x+6)\end{align}$
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d) $32x^3-80x^2-18x+45$
a
$(4x-3)(4x+3)(2x-5)$
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$\begin{align}32x^3-80x^2-18x+45&=16x^2(2x-5)-9(2x-5)\\&=(16x^2-9)(2x-5)\\&=(4x-3)(4x+3)(2x-5)\end{align}$
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Question 9
Factorise the following in pairs
a) $ax+bx+ay+by+az+bz$
a
$(x+y+z)(a+b)$
Question ID: 20210090010
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$\begin{align}ax+bx+ay+by+az+bz&=x(a+b)+y(a+b)+z(a+b)\\&=(x+y+z)(a+b)\end{align}$
Question ID: 20210090010
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b) $x^3-4x^2+x^2-4x+3x-12$
a
$(x^2+x+3)(x-4)$
Question ID: 20210090020
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$\begin{align}x^3-4x^2+x^2-4x+3x-12&=x^2(x-4)+x(x-4)+3(x-4)\\&=(x^2+x+3)(x-4)\end{align}$
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c) $6ax-3bx-2ay+by+8az-4bz$
a
$(3x-y+4z)(2a-b)$
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$\begin{align}6ax-3bx-2ay+by+8az-4bz&=3x(2a-b)-y(2a-b)+4z(2a-b)\\&=(3x-y+4z)(2a-b)\end{align}$
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d) $8x^3-16x^2-20x^2+40x-12x+24$
a
$4(2x^2-5x-3)(x-2)$
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$\begin{align}8x^3-16x^2-20x^2+40x-12x+24&=4(2x^3-4x^2-5x^2+10x-3x+6)\\&=4(2x^2(x-2)-5x(x-2)-3(x-2))\\&=4(2x^2-5x-3)(x-2)\end{align}$
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Question 10
Factorise the following in triples!
a) $ax+bx+cx+ay+by+cy$
a
$(x+y)(a+b+c)$
Question ID: 20210100010
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$\begin{align}ax+bx+cx+ay+by+cy&=x(a+b+c)+y(a+b+c)\\&=(x+y)(a+b+c)\end{align}$
Question ID: 20210100010
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b) $x^3+x^2y+2x^2-xy^2-y^2-2y^2$
a
$(x-y)(x+y)(x+y+2)$
Question ID: 20210100020
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$\begin{align}x^3+x^2y+2x^2-xy^2-y^2-2y^2&=x^2(x+y+2)-y^2(x+y+2)\\&=(x^2-y^2)(x+y+2)\\&=(x-y)(x+y)(x+y+2)\end{align}$
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Question 1
Expand the following
a) $(a+b)(x+y)$
b) $(a+2)(b+3)$
c) $(x+5)(x+8)$
d) $(x^2+x)(x+10)$
e) $(a-x)(b+y)$
f) $(x-6)(y-9)$
Question 2
Factorise the following by grouping (in pairs)
a) $px+py+qx+qy$
b) $sp+sq+tp+tq$
c) $ab+5a+3b+15$
d) $xy+10x+6y+60$
e) $x^2+8x+3x+24$
f) $y^2+7y+5y+35$
Question 3
Factorise the following by grouping (in pairs)
a) $ax+ay-bx-by$
b) $hm+hn-5m-5n$
c) $cx-2c+dx-2d$
d) $py-pz+2y-2z$
e) $x^2+7x-2x-14$
f) $x^2-x+7x-7$
Question 4
Factorise the following by grouping (in pairs)
a) $ac-ad-bc+bd$
b) $hx-2x-hy+2y$
c) $xy-x^2-6y+6x$
d) $5v^2-hv^2-5w+hw$
e) $x^2-4x-6x+24$
f) $z^2-8z-z+8$
Question 5
Factorise the following by grouping (in pairs)
a) $xy-2x+5y-10$
b) $px-py-qx+qy$
c) $3x^2+15x+4x+20$
d) $2x^2+12x-5x-30$
e) $4ap-4bp+5a-5b$
f) $5t^2-5t-6t+6$
g) $6xy-3x-8y+4$
h) $40z^2-8z-5z+1$
Question 6
Factorise the following by grouping (in pairs)
a) $20pq+24p+15q+18$
b) $9t^3+27t^2-7t-21$
c) $4x+12-x^3-3x^2$
d) $2x^3-9x^2-8x+36$
e) $3x^4-15x^2+x^2-5$
f) $20z^6-8z^3-5z^3+2$
Question 7
Factorise the following by grouping (in pairs)
a) $2xy+10x+6y+30$
b) $3x^2-27x+6x-54$
c) $2x^3+8x^2+5x^2+20x$
d) $12xyz-12xy-9xz+9x$
Question 8
Factorise the following by grouping (in pairs)
a) $x^3+2x^2-25x+2$
b) $x^3+5x^2-8x-20$
c) $9x^3+54x^2-x-6$
d) $32x^3-80x^2-18x+45$
Question 9
Factorise the following in pairs
a) $ax+bx+ay+by+az+bz$
b) $x^3-4x^2+x^2-4x+3x-12$
c) $6ax-3bx-2ay+by+8az-4bz$
d) $8x^3-16x^2-20x^2+40x-12x+24$
Question 10
Factorise the following in triples!
a) $ax+bx+cx+ay+by+cy$
b) $x^3+x^2y+2x^2-xy^2-y^2-2y^2$
Answers
Question 1
a) $ax+ay+bx+by$
b) $ab+3a+2b+6$
c) $x^2+13x+40$
d) $x^3+11x^2+10x$
e) $ab+ay-bx-xy$
f) $xy-9x-6y+54$
Question 2
a) $(p+q)(x+y)$
b) $(s+t)(p+q)$
c) $(a+3)(b+5)$
d) $(x+6)(y+10)$
e) $(x+3)(x+8)$
f) $(y+5)(y+7)$
Question 3
a) $(a-b)(x+y)$
b) $(h-5)(m+n)$
c) $(c+d)(x-2)$
d) $(p+2)(y-z)$
e) $(x-2)(x+7)$
f) $(x+7)(x-1)$
Question 4
a) $(a-b)(c-d)$
b) $(x-y)(h-2)$
c) $(x-6)(y-x)$
d) $(v^2-w)(5-h)$
e) $(x-6)(x-4)$
f) $(z-1)(z-8)$
Question 5
a) $(x+5)(y-2)$
b) $(p-q)(x-y)$
c) $(3x+4)(x+5)$
d) $(2x-5)(x+6)$
e) $(4p+5)(a-b)$
f) $(5t-6)(t-1)$
g) $(3x-4)(2y-1)$
h) $(8z-1)(5z-1)$
Question 6
a) $(4p+3)(5q+6)$
b) $(9t^2-7)(t+3)$
c) $(4-x^2)(x+3)$
d) $(x^2-4)(2x-9)$
e) $(3x^2+1)(x^2-5)$
f) $(4z^3-1)(5z^3-2)$
Question 7
a) $2(x+3)(y+5)$
b) $3(x+2)(x-9)$
c) $x(2x+5)(x+4)$
d) $3x(4y-3)(z-1)$
Question 8
a) $(x-5)(x+5)(x+2)$
b) $(x-2)(x+2)(2x+5)$
c) $(3x-1)(3x+1)(x+6)$
d) $(4x-3)(4x+3)(2x-5)$
Question 9
a) $(x+y+z)(a+b)$
b) $(x^2+x+3)(x-4)$
c) $(3x-y+4z)(2a-b)$
d) $4(2x^2-5x-3)(x-2)$
Question 10
a) $(x+y)(a+b+c)$
b) $(x-y)(x+y)(x+y+2)$
Question 1
Expand the following
a) $(a+b)(x+y)$
b) $(a+2)(b+3)$
c) $(x+5)(x+8)$
d) $(x^2+x)(x+10)$
e) $(a-x)(b+y)$
f) $(x-6)(y-9)$
Question 2
Factorise the following by grouping (in pairs)
a) $px+py+qx+qy$
b) $sp+sq+tp+tq$
c) $ab+5a+3b+15$
d) $xy+10x+6y+60$
e) $x^2+8x+3x+24$
f) $y^2+7y+5y+35$
Question 3
Factorise the following by grouping (in pairs)
a) $ax+ay-bx-by$
b) $hm+hn-5m-5n$
c) $cx-2c+dx-2d$
d) $py-pz+2y-2z$
e) $x^2+7x-2x-14$
f) $x^2-x+7x-7$
Question 4
Factorise the following by grouping (in pairs)
a) $ac-ad-bc+bd$
b) $hx-2x-hy+2y$
c) $xy-x^2-6y+6x$
d) $5v^2-hv^2-5w+hw$
e) $x^2-4x-6x+24$
f) $z^2-8z-z+8$
Question 5
Factorise the following by grouping (in pairs)
a) $xy-2x+5y-10$
b) $px-py-qx+qy$
c) $3x^2+15x+4x+20$
d) $2x^2+12x-5x-30$
e) $4ap-4bp+5a-5b$
f) $5t^2-5t-6t+6$
g) $6xy-3x-8y+4$
h) $40z^2-8z-5z+1$
Question 6
Factorise the following by grouping (in pairs)
a) $20pq+24p+15q+18$
b) $9t^3+27t^2-7t-21$
c) $4x+12-x^3-3x^2$
d) $2x^3-9x^2-8x+36$
e) $3x^4-15x^2+x^2-5$
f) $20z^6-8z^3-5z^3+2$
Question 7
Factorise the following by grouping (in pairs)
a) $2xy+10x+6y+30$
b) $3x^2-27x+6x-54$
c) $2x^3+8x^2+5x^2+20x$
d) $12xyz-12xy-9xz+9x$
Question 8
Factorise the following by grouping (in pairs)
a) $x^3+2x^2-25x+2$
b) $x^3+5x^2-8x-20$
c) $9x^3+54x^2-x-6$
d) $32x^3-80x^2-18x+45$
Question 9
Factorise the following in pairs
a) $ax+bx+ay+by+az+bz$
b) $x^3-4x^2+x^2-4x+3x-12$
c) $6ax-3bx-2ay+by+8az-4bz$
d) $8x^3-16x^2-20x^2+40x-12x+24$
Question 10
Factorise the following in triples!
a) $ax+bx+cx+ay+by+cy$
b) $x^3+x^2y+2x^2-xy^2-y^2-2y^2$
Answers
Question 1
a) $ax+ay+bx+by$
b) $ab+3a+2b+6$
c) $x^2+13x+40$
d) $x^3+11x^2+10x$
e) $ab+ay-bx-xy$
f) $xy-9x-6y+54$
Question 2
a) $(p+q)(x+y)$
b) $(s+t)(p+q)$
c) $(a+3)(b+5)$
d) $(x+6)(y+10)$
e) $(x+3)(x+8)$
f) $(y+5)(y+7)$
Question 3
a) $(a-b)(x+y)$
b) $(h-5)(m+n)$
c) $(c+d)(x-2)$
d) $(p+2)(y-z)$
e) $(x-2)(x+7)$
f) $(x+7)(x-1)$
Question 4
a) $(a-b)(c-d)$
b) $(x-y)(h-2)$
c) $(x-6)(y-x)$
d) $(v^2-w)(5-h)$
e) $(x-6)(x-4)$
f) $(z-1)(z-8)$
Question 5
a) $(x+5)(y-2)$
b) $(p-q)(x-y)$
c) $(3x+4)(x+5)$
d) $(2x-5)(x+6)$
e) $(4p+5)(a-b)$
f) $(5t-6)(t-1)$
g) $(3x-4)(2y-1)$
h) $(8z-1)(5z-1)$
Question 6
a) $(4p+3)(5q+6)$
b) $(9t^2-7)(t+3)$
c) $(4-x^2)(x+3)$
d) $(x^2-4)(2x-9)$
e) $(3x^2+1)(x^2-5)$
f) $(4z^3-1)(5z^3-2)$
Question 7
a) $2(x+3)(y+5)$
b) $3(x+2)(x-9)$
c) $x(2x+5)(x+4)$
d) $3x(4y-3)(z-1)$
Question 8
a) $(x-5)(x+5)(x+2)$
b) $(x-2)(x+2)(2x+5)$
c) $(3x-1)(3x+1)(x+6)$
d) $(4x-3)(4x+3)(2x-5)$
Question 9
a) $(x+y+z)(a+b)$
b) $(x^2+x+3)(x-4)$
c) $(3x-y+4z)(2a-b)$
d) $4(2x^2-5x-3)(x-2)$
Question 10
a) $(x+y)(a+b+c)$
b) $(x-y)(x+y)(x+y+2)$
Solutions
Question 1
a) $\begin{align}(a+b)(x+y)&=a(x+y)+b(x+y)\\&=ax+ay+bx+by\end{align}$
b) $\begin{align}(a+2)(b+3)&=a(b+3)+2(b+3)\\&=ab+3a+2b+6\end{align}$
c) $\begin{align}(x+5)(x+8)&=x(x+8)+5(x+8)\\&=x^2+8x+5x+40\\&=x^2+13x+40\end{align}$
d) $\begin{align}(x^2+x)(x+10)&=x^2(x+10)+x(x+10)\\&=x^3+10x^2+x^2+10x\\&=x^3+11x^2+10x\end{align}$
e) $\begin{align}(a-x)(b+y)&=a(b+y)-x(b+y)\\&=ab+ay-bx-xy\end{align}$
f) $\begin{align}(x-6)(y-9)&=x(y-9)-6(y-9)\\&=xy-9x-6y+54\end{align}$
Question 2
a) $\begin{align}px+py+qx+qy&=p(x+y)+q(x+y)\\&=(p+q)(x+y)\end{align}$
b) $\begin{align}sp+sq+tp+tq&=s(p+q)+t(p+q)\\&=(s+t)(p+q)\end{align}$
c) $\begin{align}ab+5a+3b+15&=a(b+5)+3(b+5)\\&=(a+3)(b+5)\end{align}$
d) $\begin{align}xy+10x+6y+60&=x(y+10)+6(y+10)\\&=(x+6)(y+10)\end{align}$
e) $\begin{align}x^2+8x+3x+24&=x(x+8)+3(x+8)\\&=(x+3)(x+8)\end{align}$
f) $\begin{align}y^2+7y+5y+35&=y(y+7)+5(y+7)\\&=(y+5)(y+7)\end{align}$
Question 3
a) $\begin{align}ax+ay-bx-by&=a(x+y)-b(x+y)\\&=(a-b)(x+y)\end{align}$
b) $\begin{align}hm+hn-5m-5n&=h(m+n)-5(m+n)\\&=(h-5)(m+n)\end{align}$
c) $\begin{align}cx-2c+dx-2d&=c(x-2)+d(x-2)\\&=(c+d)(x-2)\end{align}$
d) $\begin{align}py-pz+2y-2z&=p(y-z)+2(y-z)\\&=(p+2)(y-z)\end{align}$
e) $\begin{align}x^2+7x-2x-14&=x(x+7)-2(x+7)\\&=(x-2)(x+7)\end{align}$
f) $\begin{align}x^2-x+7x-7&=x(x-1)+7(x-1)\\&=(x+7)(x-1)\end{align}$
Question 4
a) $\begin{align}ac-ad-bc+bd&=a(c-d)-b(c-d)\\&=(a-b)(c-d)\end{align}$
b) $\begin{align}hx-2x-hy+2y&=x(h-2)-y(h-2)\\&=(x-y)(h-2)\end{align}$
c) $\begin{align}xy-x^2-6y+6x&=x(y-x)-6(y-x)\\&=(x-6)(y-x)\end{align}$
d) $\begin{align}5v^2-hv^2-5w+hw&=v^2(5-h)-w(5-h)\\&=(v^2-w)(5-h)\end{align}$
e) $\begin{align}x^2-4x-6x+24&=x(x-4)-6(x-4)\\&=(x-6)(x-4)\end{align}$
f) $\begin{align}z^2-8z-z+8&=z(z-8)-1(z-8)\\&=(z-1)(z-8)\end{align}$
Question 5
a) $\begin{align}xy-2x+5y-10&=x(y-2)+5(y-2)\\&=(x+5)(y-2)\end{align}$
b) $\begin{align}px-py-qx+qy&=p(x-y)-q(x-y)\\&=(p-q)(x-y)\end{align}$
c) $\begin{align}3x^2+15x+4x+20&=3x(x+5)+4(x+5)\\&=(3x+4)(x+5)\end{align}$
d) $\begin{align}2x^2+12x-5x-30&=2x(x+6)-5(x+6)\\&=(2x-5)(x+6)\end{align}$
e) $\begin{align}4ap-4bp+5a-5b&=4p(a-b)+5(a-b)\\&=(4p+5)(a-b)\end{align}$
f) $\begin{align}5t^2-5t-6t+6&=5t(t-1)-6(t-1)\\&=(5t-6)(t-1)\end{align}$
g) $\begin{align}6xy-3x-8y+4&=3x(2y-1)-4(2y-1)\\&=(3x-4)(2y-1)\end{align}$
h) $\begin{align}40z^2-8z-5z+1&=8z(5z-1)-1(5z-1)\\&=(8z-1)(5z-1)\end{align}$
Question 6
a) $\begin{align}20pq+24p+15q+18&=4p(5q+6)+3(5q+6)\\&=(4p+3)(5q+6)\end{align}$
b) $\begin{align}9t^3+27t^2-7t-21&=9t^2(t+3)-7(t+3)\\&=(9t^2-7)(t+3)\end{align}$
c) $\begin{align}4x+12-x^3-3x^2&=4(x+3)-x^2(x+3)\\&=(4-x^2)(x+3)\end{align}$
d) $\begin{align}2x^3-9x^2-8x+36&=x^2(2x-9)-4(2x-9)\\&=(x^2-4)(2x-9)\end{align}$
e) $\begin{align}3x^4-15x^2+x^2-5&=3x^2(x^2-5)+1(x^2-5)\\&=(3x^2+1)(x^2-5)\end{align}$
f) $\begin{align}20z^6-8z^3-5z^3+2&=4z^3(5z^3-2)-1(5z^3-2)\\&=(4z^3-1)(5z^3-2)\end{align}$
Question 7
a) $\begin{align}2xy+10x+6y+30&=2(xy+5x+3y+15)\\&=2(x(y+5)+3(y+5))\\&=2(x+3)(y+5)\end{align}$
b) $\begin{align}3x^2-27x+6x-54&=3(x^2-9x+2x-18)\\&=3(x(x-9)+2(x-9))\\&=3(x+2)(x-9)\end{align}$
c) $\begin{align}2x^3+8x^2+5x^2+20x&=x(2x^2+8x+5x+20)\\&=x(2x(x+4)+5(x+4))\\&=x(2x+5)(x+4)\end{align}$
d) $\begin{align}12xyz-12xy-9xz+9x&=3x(4yz-4y-3z+3)\\&=3x(4y(z-1)-3(z-1))\\&=3x(4y-3)(z-1)\end{align}$
Question 8
a) $\begin{align}x^3+2x^2-25x+2&=x^2(x+2)-25(x+2)\\&=(x^2-25)(x+2)\\&=(x-5)(x+5)(x+2)\end{align}$
b) $\begin{align}x^3+5x^2-8x-20&=x^2(2x+5)-4(2x+5)\\&=(x^2-4)(2x+5)\\&=(x-2)(x+2)(2x+5)\end{align}$
c) $\begin{align}9x^3+54x^2-x-6&=9x^2(x+6)-1(x+6)\\&=(9x^2-1)(x+6)\\&=(3x-1)(3x+1)(x+6)\end{align}$
d) $\begin{align}32x^3-80x^2-18x+45&=16x^2(2x-5)-9(2x-5)\\&=(16x^2-9)(2x-5)\\&=(4x-3)(4x+3)(2x-5)\end{align}$
Question 9
a) $\begin{align}ax+bx+ay+by+az+bz&=x(a+b)+y(a+b)+z(a+b)\\&=(x+y+z)(a+b)\end{align}$
b) $\begin{align}x^3-4x^2+x^2-4x+3x-12&=x^2(x-4)+x(x-4)+3(x-4)\\&=(x^2+x+3)(x-4)\end{align}$
c) $\begin{align}6ax-3bx-2ay+by+8az-4bz&=3x(2a-b)-y(2a-b)+4z(2a-b)\\&=(3x-y+4z)(2a-b)\end{align}$
d) $\begin{align}8x^3-16x^2-20x^2+40x-12x+24&=4(2x^3-4x^2-5x^2+10x-3x+6)\\&=4(2x^2(x-2)-5x(x-2)-3(x-2))\\&=4(2x^2-5x-3)(x-2)\end{align}$
Question 10
a) $\begin{align}ax+bx+cx+ay+by+cy&=x(a+b+c)+y(a+b+c)\\&=(x+y)(a+b+c)\end{align}$
b) $\begin{align}x^3+x^2y+2x^2-xy^2-y^2-2y^2&=x^2(x+y+2)-y^2(x+y+2)\\&=(x^2-y^2)(x+y+2)\\&=(x-y)(x+y)(x+y+2)\end{align}$