Algebra - The Difference Of Two Squares
Question 1
Expand, notice any patterns:
a) $(x+4)(x-4)$
a
$x^2-16$
Question ID: 20190010010
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b) $(x+8)(x-8)$
a
$x^2-64$
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c) $(x-3)(x+3)$
a
$x^2-9$
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d) $(z-10)(z+10)$
a
$z^2-100$
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e) $(9+x)(9-x)$
a
$81-x^2$
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f) $(2x+3)(2x-3)$
a
$4x^2-9$
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g) $(5-3y)(5+3y)$
a
$25-9y^2$
Question ID: 20190010070
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h) $2(x+6)(x-6)$
a
$2x^2-72$
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i) $(4z+3x)(4z-3x)$
a
$16z^2-9x^2$
Question ID: 20190010090
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j) $(x^n+2)(x^n-2)$
a
$x^{2n}-4$
Question ID: 20190010100
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k) $4x(5-6x)(5+6x)$
a
$100x-144x^3$
Question ID: 20190010110
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l) $(x+y+11)(x+y-11)$
a
$(x+y)^2-121$
Question ID: 20190010120
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Question 2
Factorise the following using the difference of two squares:
a) $x^2-25$
a
$(x+5)(x-5)$
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b) $x^2-49$
a
$(x+7)(x-7)$
Question ID: 20190020020
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c) $x^2-81$
a
$(x+9)(x-9)$
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d) $x^2-144$
a
$(x+12)(x-12)$
Question ID: 20190020040
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e) $x^2-1$
a
$(x+1)(x-1)$
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f) $z^2-4$
a
$(z+2)(z-2)$
Question ID: 20190020060
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g) $y^2-100$
a
$(y+10)(y-10)$
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h) $a^2-16$
a
$(a+4)(a-4)$
Question ID: 20190020080
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Question 3
By taking out a common factor first, factorise the following:
a) $2x^2-18$
a
$2(x+3)(x-3)$
Question ID: 20190030010
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b) $3x^2-75$
a
$3(x+5)(x-5)$
Question ID: 20190030020
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c) $5x^2-500$
a
$5(x+10)(x-10)$
Question ID: 20190030030
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d) $x^3-64x$
a
$x(x+8)(x-8)$
Question ID: 20190030040
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e) $x^4-36x^2$
a
$x^2(x+6)(x-6)$
Question ID: 20190030050
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f) $2x^3-8x$
a
$2x(x+2)(x-2)$
Question ID: 20190030060
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g) $7x^3-7x$
a
$7x(x+1)(x-1)$
Question ID: 20190030070
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h) $9x^4-144x^2$
a
$9x^2(x+4)(x-4)$
Question ID: 20190030080
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Question 4
Factorise:
a) $4x^2-9$
a
$(2x-3)(2x+3)$
Question ID: 20190040010
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b) $9x^2-4$
a
$(3x-2)(3x+2)$
Question ID: 20190040020
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c) $25x^2-16$
a
$(5x-4)(5x+4)$
Question ID: 20190040030
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d) $16x^2-1$
a
$(4x-1)(4x+1)$
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e) $49x^2-25$
a
$(7x-5)(7x+5)$
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f) $x^4-9$
a
$(x^2-3)(x^2+3)$
Question ID: 20190040060
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g) $x^6-1$
a
$(x^3-1)(x^3+1)$
Question ID: 20190040070
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h) $x^{10}-49$
a
$(x^5-7)(x^5+7)$
Question ID: 20190040080
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i) $9x^4-25$
a
$(3x^2-5)(3x^2+5)$
Question ID: 20190040090
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j) $25x^8-81$
a
$(5x^4-9)(5x^4+9)$
Question ID: 20190040100
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Question 5
Factorise the following, you may need to take out a common factor first:
a) $9x^3-16x$
a
$x(3x-4)(3x+4)$
Question ID: 20190050010
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b) $16z^3-z$
a
$z(4z-1)(4z+1)$
Question ID: 20190050020
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c) $9-4x^2$
a
$(3-2x)(3+2x)$
Question ID: 20190050030
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d) $64-25x^2$
a
$(8-5x)(8+5x)$
Question ID: 20190050040
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e) $18x^2-2$
a
$2(3x+1)(3x-1)$
Question ID: 20190050050
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f) $125x^2-20$
a
$5(5x+2)(5x-2)$
Question ID: 20190050060
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g) $36x^3-100x$
a
$4x(3x+5)(3x-5)$
Question ID: 20190050070
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h) $12x-27x^3$
a
$3x(2-3x)(2+3x)$
Question ID: 20190050080
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Question 6
Factorise:
a) $y^2-z^2$
a
$(y+z)(y-z)$
Question ID: 20190060010
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b) $a^2-b^2$
a
$(a+b)(a-b)$
Question ID: 20190060020
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c) $4x^2-y^2$
a
$(2x-y)(2x+y)$
Question ID: 20190060030
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d) $9y^2-4x^2$
a
$(3y-2x)(3y+2x)$
Question ID: 20190060040
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e) $2x^2-18z^2$
a
$2(x-3z)(x+3z)$
Question ID: 20190060050
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f) $x^2z-y^2z$
a
$z(x-y)(x+y)$
Question ID: 20190060060
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g) $48x^2-75y^2$
a
$3(4x+5y)(4x-5y)$
Question ID: 20190060070
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h) $45x^3-80xy^2$
a
$5x(3x+4y)(3x-4y)$
Question ID: 20190060080
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Question 7
Factorise:
a) $x^2-\frac19$
a
$\left(x-\frac13\right)\left(x+\frac13\right)$
Question ID: 20190065010
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b) $y^2-\frac49$
a
$\left(y-\frac23\right)\left(y+\frac23\right)$
Question ID: 20190065020
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c) $9x^2-\frac14$
a
$\left(3x-\frac12\right)\left(3x+\frac12\right)$
Question ID: 20190065030
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d) $36x^2-\frac{25}{49}$
a
$\left(6x-\frac57\right)\left(6x+\frac57\right)$
Question ID: 20190065040
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e) $\frac14x^2-\frac{1}{16}$
a
$\left(\frac12x-\frac14\right)\left(\frac12x+\frac14\right)$
Question ID: 20190065050
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f) $\frac {x^2}{9}-\frac{4}{25}$
a
$\left(\frac x3-\frac25\right)\left(\frac x3+\frac25\right)$
Question ID: 20190065060
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g) $\frac {9x^2}{4}-\frac{36y^2}{100}$
a
$\left(\frac{3x}{2}-\frac{6y}{10}\right)\left(\frac{3x}{2}+\frac{6y}{10}\right)$
Question ID: 20190065070
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h) $\frac {125}{9}a^2-\frac{405}{64}b^2$
a
$5\left(\frac{5}{3}a-\frac{9}{8}b\right)\left(\frac{5}{3}a-\frac{9}{8}b\right)$
Question ID: 20190065080
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Question 8
Expand, leaving answers in the form $ax^2+bx+c$:
a) $((x+2)+3)((x+2)-3)$
a
$x^2+4x-5$
Question ID: 20190070010
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$\begin{align}((x+2)+3)((x+2)-3)&=(x+2)^2-9\\&=x^2+4x-5\end{align}$
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b) $((x+1)+5)((x+1)-5)$
a
$x^2+2x-24$
Question ID: 20190070020
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$\begin{align}((x+1)+5)((x+1)-5)&=(x+1)^2-25\\&=x^2+2x-24\end{align}$
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c) $((x+3)+8)((x+2)-8)$
a
$x^2+6x-55$
Question ID: 20190070030
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$\begin{align}((x+3)+8)((x+2)-8)&=(x+3)^2-64\\&=x^2+6x-55\end{align}$
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d) $(x+y-5)(x+y+5)$
a
$(x+y)^2-25$
Question ID: 20190070040
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e) $(5+2x+y)(5-(2x+y))$
a
$25-(2x+y)^2$
Question ID: 20190070050
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f) $(9+x+y)(9-x-y)$
a
$81-(x+y)^2$
Question ID: 20190070060
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Question 9
Factorise completely:
a) $x^4-1$
a
$(x-1)(x+1)(x^2+1)$
Question ID: 20190080010
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$\begin{align}x^4-1&=(x^2-1)(x^2+1)\\&=(x-1)(x+1)(x^2+1)\end{align}$
Question ID: 20190080010
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b) $16x^4-1$
a
$(2x-1)(2x+1)(4x^2+1)$
Question ID: 20190080020
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$\begin{align}16x^4-1&=(4x^2-1)(4x^2+1)\\&=(2x-1)(2x+1)(4x^2+1)\end{align}$
Question ID: 20190080020
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c) $x^8-256$
a
$(x-2)(x+2)(x^2+4)(x^4+16)$
Question ID: 20190080030
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$\begin{align}x^8-256&=(x^4-16)(x^4+16)\\&=(x^2-4)(x^2+4)(x^4+16)\\&=(x-2)(x+2)(x^2+4)(x^4+16)\end{align}$
Question ID: 20190080030
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d) $243x^4-48$
a
$3(3x-2)(3x+2)(9x^2+4)$
Question ID: 20190080040
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$\begin{align}243x^4-48&=3(81x^4-16)\\&=3(9x^2-4)(9x^2+4)\\&=3(3x-2)(3x+2)(9x^2+4)\end{align}$
Question ID: 20190080040
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Question 10
Use the difference of two squares and irrational numbers to factorise the following:
a) $x^2-2$
a
$\left(x-\sqrt2\right)\left(x+\sqrt2\right)$
Question ID: 20190085010
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b) $x^2-5$
a
$\left(x-\sqrt{5}\right)\left(x+\sqrt{5}\right)$
Question ID: 20190085020
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c) $x^2-8$
a
$\left(x-2\sqrt{2}\right)\left(x+2\sqrt{2}\right)$
Question ID: 20190085030
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d) $y^2-20$
a
$\left(y-2\sqrt{5}\right)\left(y+2\sqrt{5}\right)$
Question ID: 20190085040
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e) $a^6-150$
a
$\left(a^3-5\sqrt{6}\right)\left(a^3+5\sqrt{6}\right)$
Question ID: 20190085050
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f) $9p^{10}-98q^2$
a
$\left(3p^5-7\sqrt{2}q\right)\left(3p^5+7\sqrt{2}q\right)$
Question ID: 20190085060
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Question 11
Factorise and simplify fully:
a) $(x+y)^2-9$
a
$(x+y+3)(x+y-3)$
Question ID: 20190090010
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b) $(x-3)^2-25$
a
$(x+2)(x-8)$
Question ID: 20190090020
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c) $(2x-1)^2-100$
a
$(2x+9)(2x-11)$
Question ID: 20190090030
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d) $(5x-2)^2-9x^2$
a
$4(4x-1)(x-1)$
Question ID: 20190090040
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e) $16x^2-(10x-3)^2$
a
$3(14x-3)(1-2x)$
Question ID: 20190090050
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f) $(4x-1)^2-(3x-5)^2$
a
$(7x-6)(x+4)$
Question ID: 20190090060
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g) $25(x+5)^2-100(2x-3)^2$
a
$25(5x-1)(11-3x)$
Question ID: 20190090070
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$\begin{align}25(x+5)^2-100(2x-3)^2&=(5(x+5)+10(2x-3))(5(x+5)-10(2x-3))\\&=(5x+25+20x-30)(5x+25-20x+30)\\&=(25x-5)(55-15x)\\&=5(5x-1)5(11-3x)\\&=25(5x-1)(11-3x)\end{align}$
Question ID: 20190090070
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h) $49(3x-2)^4-4(7x-1)^2$
a
$(63x^2-70x+26)(63x^2-98x+30)$
Question ID: 20190090080
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$\begin{align}49(3x-2)^4-4(7x-1)^2&=(7(3x-2)^2+2(7x-1))(7(3x-2)^2-2(7x-1))\\&=(7(9x^2-12x+4)+14x-2)(7(9x^2-12x+4)-14x+2)\\&=(63x^2-84x+28+14x-2)(63x^2-84x+28-14x+2)\\&=(63x^2-70x+26)(63x^2-98x+30)\end{align}$
Question ID: 20190090080
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Question 1
Expand, notice any patterns:
a) $(x+4)(x-4)$
b) $(x+8)(x-8)$
c) $(x-3)(x+3)$
d) $(z-10)(z+10)$
e) $(9+x)(9-x)$
f) $(2x+3)(2x-3)$
g) $(5-3y)(5+3y)$
h) $2(x+6)(x-6)$
i) $(4z+3x)(4z-3x)$
j) $(x^n+2)(x^n-2)$
k) $4x(5-6x)(5+6x)$
l) $(x+y+11)(x+y-11)$
Question 2
Factorise the following using the difference of two squares:
a) $x^2-25$
b) $x^2-49$
c) $x^2-81$
d) $x^2-144$
e) $x^2-1$
f) $z^2-4$
g) $y^2-100$
h) $a^2-16$
Question 3
By taking out a common factor first, factorise the following:
a) $2x^2-18$
b) $3x^2-75$
c) $5x^2-500$
d) $x^3-64x$
e) $x^4-36x^2$
f) $2x^3-8x$
g) $7x^3-7x$
h) $9x^4-144x^2$
Question 4
Factorise:
a) $4x^2-9$
b) $9x^2-4$
c) $25x^2-16$
d) $16x^2-1$
e) $49x^2-25$
f) $x^4-9$
g) $x^6-1$
h) $x^{10}-49$
i) $9x^4-25$
j) $25x^8-81$
Question 5
Factorise the following, you may need to take out a common factor first:
a) $9x^3-16x$
b) $16z^3-z$
c) $9-4x^2$
d) $64-25x^2$
e) $18x^2-2$
f) $125x^2-20$
g) $36x^3-100x$
h) $12x-27x^3$
Question 6
Factorise:
a) $y^2-z^2$
b) $a^2-b^2$
c) $4x^2-y^2$
d) $9y^2-4x^2$
e) $2x^2-18z^2$
f) $x^2z-y^2z$
g) $48x^2-75y^2$
h) $45x^3-80xy^2$
Question 7
Factorise:
a) $x^2-\frac19$
b) $y^2-\frac49$
c) $9x^2-\frac14$
d) $36x^2-\frac{25}{49}$
e) $\frac14x^2-\frac{1}{16}$
f) $\frac {x^2}{9}-\frac{4}{25}$
g) $\frac {9x^2}{4}-\frac{36y^2}{100}$
h) $\frac {125}{9}a^2-\frac{405}{64}b^2$
Question 8
Expand, leaving answers in the form $ax^2+bx+c$:
a) $((x+2)+3)((x+2)-3)$
b) $((x+1)+5)((x+1)-5)$
c) $((x+3)+8)((x+2)-8)$
d) $(x+y-5)(x+y+5)$
e) $(5+2x+y)(5-(2x+y))$
f) $(9+x+y)(9-x-y)$
Question 9
Factorise completely:
a) $x^4-1$
b) $16x^4-1$
c) $x^8-256$
d) $243x^4-48$
Question 10
Use the difference of two squares and irrational numbers to factorise the following:
a) $x^2-2$
b) $x^2-5$
c) $x^2-8$
d) $y^2-20$
e) $a^6-150$
f) $9p^{10}-98q^2$
Question 11
Factorise and simplify fully:
a) $(x+y)^2-9$
b) $(x-3)^2-25$
c) $(2x-1)^2-100$
d) $(5x-2)^2-9x^2$
e) $16x^2-(10x-3)^2$
f) $(4x-1)^2-(3x-5)^2$
g) $25(x+5)^2-100(2x-3)^2$
h) $49(3x-2)^4-4(7x-1)^2$
Answers
Question 1
a) $x^2-16$
b) $x^2-64$
c) $x^2-9$
d) $z^2-100$
e) $81-x^2$
f) $4x^2-9$
g) $25-9y^2$
h) $2x^2-72$
i) $16z^2-9x^2$
j) $x^{2n}-4$
k) $100x-144x^3$
l) $(x+y)^2-121$
Question 2
a) $(x+5)(x-5)$
b) $(x+7)(x-7)$
c) $(x+9)(x-9)$
d) $(x+12)(x-12)$
e) $(x+1)(x-1)$
f) $(z+2)(z-2)$
g) $(y+10)(y-10)$
h) $(a+4)(a-4)$
Question 3
a) $2(x+3)(x-3)$
b) $3(x+5)(x-5)$
c) $5(x+10)(x-10)$
d) $x(x+8)(x-8)$
e) $x^2(x+6)(x-6)$
f) $2x(x+2)(x-2)$
g) $7x(x+1)(x-1)$
h) $9x^2(x+4)(x-4)$
Question 4
a) $(2x-3)(2x+3)$
b) $(3x-2)(3x+2)$
c) $(5x-4)(5x+4)$
d) $(4x-1)(4x+1)$
e) $(7x-5)(7x+5)$
f) $(x^2-3)(x^2+3)$
g) $(x^3-1)(x^3+1)$
h) $(x^5-7)(x^5+7)$
i) $(3x^2-5)(3x^2+5)$
j) $(5x^4-9)(5x^4+9)$
Question 5
a) $x(3x-4)(3x+4)$
b) $z(4z-1)(4z+1)$
c) $(3-2x)(3+2x)$
d) $(8-5x)(8+5x)$
e) $2(3x+1)(3x-1)$
f) $5(5x+2)(5x-2)$
g) $4x(3x+5)(3x-5)$
h) $3x(2-3x)(2+3x)$
Question 6
a) $(y+z)(y-z)$
b) $(a+b)(a-b)$
c) $(2x-y)(2x+y)$
d) $(3y-2x)(3y+2x)$
e) $2(x-3z)(x+3z)$
f) $z(x-y)(x+y)$
g) $3(4x+5y)(4x-5y)$
h) $5x(3x+4y)(3x-4y)$
Question 7
a) $\left(x-\frac13\right)\left(x+\frac13\right)$
b) $\left(y-\frac23\right)\left(y+\frac23\right)$
c) $\left(3x-\frac12\right)\left(3x+\frac12\right)$
d) $\left(6x-\frac57\right)\left(6x+\frac57\right)$
e) $\left(\frac12x-\frac14\right)\left(\frac12x+\frac14\right)$
f) $\left(\frac x3-\frac25\right)\left(\frac x3+\frac25\right)$
g) $\left(\frac{3x}{2}-\frac{6y}{10}\right)\left(\frac{3x}{2}+\frac{6y}{10}\right)$
h) $5\left(\frac{5}{3}a-\frac{9}{8}b\right)\left(\frac{5}{3}a-\frac{9}{8}b\right)$
Question 8
a) $x^2+4x-5$
b) $x^2+2x-24$
c) $x^2+6x-55$
d) $(x+y)^2-25$
e) $25-(2x+y)^2$
f) $81-(x+y)^2$
Question 9
a) $(x-1)(x+1)(x^2+1)$
b) $(2x-1)(2x+1)(4x^2+1)$
c) $(x-2)(x+2)(x^2+4)(x^4+16)$
d) $3(3x-2)(3x+2)(9x^2+4)$
Question 10
a) $\left(x-\sqrt2\right)\left(x+\sqrt2\right)$
b) $\left(x-\sqrt{5}\right)\left(x+\sqrt{5}\right)$
c) $\left(x-2\sqrt{2}\right)\left(x+2\sqrt{2}\right)$
d) $\left(y-2\sqrt{5}\right)\left(y+2\sqrt{5}\right)$
e) $\left(a^3-5\sqrt{6}\right)\left(a^3+5\sqrt{6}\right)$
f) $\left(3p^5-7\sqrt{2}q\right)\left(3p^5+7\sqrt{2}q\right)$
Question 11
a) $(x+y+3)(x+y-3)$
b) $(x+2)(x-8)$
c) $(2x+9)(2x-11)$
d) $4(4x-1)(x-1)$
e) $3(14x-3)(1-2x)$
f) $(7x-6)(x+4)$
g) $25(5x-1)(11-3x)$
h) $(63x^2-70x+26)(63x^2-98x+30)$
Question 1
Expand, notice any patterns:
a) $(x+4)(x-4)$
b) $(x+8)(x-8)$
c) $(x-3)(x+3)$
d) $(z-10)(z+10)$
e) $(9+x)(9-x)$
f) $(2x+3)(2x-3)$
g) $(5-3y)(5+3y)$
h) $2(x+6)(x-6)$
i) $(4z+3x)(4z-3x)$
j) $(x^n+2)(x^n-2)$
k) $4x(5-6x)(5+6x)$
l) $(x+y+11)(x+y-11)$
Question 2
Factorise the following using the difference of two squares:
a) $x^2-25$
b) $x^2-49$
c) $x^2-81$
d) $x^2-144$
e) $x^2-1$
f) $z^2-4$
g) $y^2-100$
h) $a^2-16$
Question 3
By taking out a common factor first, factorise the following:
a) $2x^2-18$
b) $3x^2-75$
c) $5x^2-500$
d) $x^3-64x$
e) $x^4-36x^2$
f) $2x^3-8x$
g) $7x^3-7x$
h) $9x^4-144x^2$
Question 4
Factorise:
a) $4x^2-9$
b) $9x^2-4$
c) $25x^2-16$
d) $16x^2-1$
e) $49x^2-25$
f) $x^4-9$
g) $x^6-1$
h) $x^{10}-49$
i) $9x^4-25$
j) $25x^8-81$
Question 5
Factorise the following, you may need to take out a common factor first:
a) $9x^3-16x$
b) $16z^3-z$
c) $9-4x^2$
d) $64-25x^2$
e) $18x^2-2$
f) $125x^2-20$
g) $36x^3-100x$
h) $12x-27x^3$
Question 6
Factorise:
a) $y^2-z^2$
b) $a^2-b^2$
c) $4x^2-y^2$
d) $9y^2-4x^2$
e) $2x^2-18z^2$
f) $x^2z-y^2z$
g) $48x^2-75y^2$
h) $45x^3-80xy^2$
Question 7
Factorise:
a) $x^2-\frac19$
b) $y^2-\frac49$
c) $9x^2-\frac14$
d) $36x^2-\frac{25}{49}$
e) $\frac14x^2-\frac{1}{16}$
f) $\frac {x^2}{9}-\frac{4}{25}$
g) $\frac {9x^2}{4}-\frac{36y^2}{100}$
h) $\frac {125}{9}a^2-\frac{405}{64}b^2$
Question 8
Expand, leaving answers in the form $ax^2+bx+c$:
a) $((x+2)+3)((x+2)-3)$
b) $((x+1)+5)((x+1)-5)$
c) $((x+3)+8)((x+2)-8)$
d) $(x+y-5)(x+y+5)$
e) $(5+2x+y)(5-(2x+y))$
f) $(9+x+y)(9-x-y)$
Question 9
Factorise completely:
a) $x^4-1$
b) $16x^4-1$
c) $x^8-256$
d) $243x^4-48$
Question 10
Use the difference of two squares and irrational numbers to factorise the following:
a) $x^2-2$
b) $x^2-5$
c) $x^2-8$
d) $y^2-20$
e) $a^6-150$
f) $9p^{10}-98q^2$
Question 11
Factorise and simplify fully:
a) $(x+y)^2-9$
b) $(x-3)^2-25$
c) $(2x-1)^2-100$
d) $(5x-2)^2-9x^2$
e) $16x^2-(10x-3)^2$
f) $(4x-1)^2-(3x-5)^2$
g) $25(x+5)^2-100(2x-3)^2$
h) $49(3x-2)^4-4(7x-1)^2$
Answers
Question 1
a) $x^2-16$
b) $x^2-64$
c) $x^2-9$
d) $z^2-100$
e) $81-x^2$
f) $4x^2-9$
g) $25-9y^2$
h) $2x^2-72$
i) $16z^2-9x^2$
j) $x^{2n}-4$
k) $100x-144x^3$
l) $(x+y)^2-121$
Question 2
a) $(x+5)(x-5)$
b) $(x+7)(x-7)$
c) $(x+9)(x-9)$
d) $(x+12)(x-12)$
e) $(x+1)(x-1)$
f) $(z+2)(z-2)$
g) $(y+10)(y-10)$
h) $(a+4)(a-4)$
Question 3
a) $2(x+3)(x-3)$
b) $3(x+5)(x-5)$
c) $5(x+10)(x-10)$
d) $x(x+8)(x-8)$
e) $x^2(x+6)(x-6)$
f) $2x(x+2)(x-2)$
g) $7x(x+1)(x-1)$
h) $9x^2(x+4)(x-4)$
Question 4
a) $(2x-3)(2x+3)$
b) $(3x-2)(3x+2)$
c) $(5x-4)(5x+4)$
d) $(4x-1)(4x+1)$
e) $(7x-5)(7x+5)$
f) $(x^2-3)(x^2+3)$
g) $(x^3-1)(x^3+1)$
h) $(x^5-7)(x^5+7)$
i) $(3x^2-5)(3x^2+5)$
j) $(5x^4-9)(5x^4+9)$
Question 5
a) $x(3x-4)(3x+4)$
b) $z(4z-1)(4z+1)$
c) $(3-2x)(3+2x)$
d) $(8-5x)(8+5x)$
e) $2(3x+1)(3x-1)$
f) $5(5x+2)(5x-2)$
g) $4x(3x+5)(3x-5)$
h) $3x(2-3x)(2+3x)$
Question 6
a) $(y+z)(y-z)$
b) $(a+b)(a-b)$
c) $(2x-y)(2x+y)$
d) $(3y-2x)(3y+2x)$
e) $2(x-3z)(x+3z)$
f) $z(x-y)(x+y)$
g) $3(4x+5y)(4x-5y)$
h) $5x(3x+4y)(3x-4y)$
Question 7
a) $\left(x-\frac13\right)\left(x+\frac13\right)$
b) $\left(y-\frac23\right)\left(y+\frac23\right)$
c) $\left(3x-\frac12\right)\left(3x+\frac12\right)$
d) $\left(6x-\frac57\right)\left(6x+\frac57\right)$
e) $\left(\frac12x-\frac14\right)\left(\frac12x+\frac14\right)$
f) $\left(\frac x3-\frac25\right)\left(\frac x3+\frac25\right)$
g) $\left(\frac{3x}{2}-\frac{6y}{10}\right)\left(\frac{3x}{2}+\frac{6y}{10}\right)$
h) $5\left(\frac{5}{3}a-\frac{9}{8}b\right)\left(\frac{5}{3}a-\frac{9}{8}b\right)$
Question 8
a) $x^2+4x-5$
b) $x^2+2x-24$
c) $x^2+6x-55$
d) $(x+y)^2-25$
e) $25-(2x+y)^2$
f) $81-(x+y)^2$
Question 9
a) $(x-1)(x+1)(x^2+1)$
b) $(2x-1)(2x+1)(4x^2+1)$
c) $(x-2)(x+2)(x^2+4)(x^4+16)$
d) $3(3x-2)(3x+2)(9x^2+4)$
Question 10
a) $\left(x-\sqrt2\right)\left(x+\sqrt2\right)$
b) $\left(x-\sqrt{5}\right)\left(x+\sqrt{5}\right)$
c) $\left(x-2\sqrt{2}\right)\left(x+2\sqrt{2}\right)$
d) $\left(y-2\sqrt{5}\right)\left(y+2\sqrt{5}\right)$
e) $\left(a^3-5\sqrt{6}\right)\left(a^3+5\sqrt{6}\right)$
f) $\left(3p^5-7\sqrt{2}q\right)\left(3p^5+7\sqrt{2}q\right)$
Question 11
a) $(x+y+3)(x+y-3)$
b) $(x+2)(x-8)$
c) $(2x+9)(2x-11)$
d) $4(4x-1)(x-1)$
e) $3(14x-3)(1-2x)$
f) $(7x-6)(x+4)$
g) $25(5x-1)(11-3x)$
h) $(63x^2-70x+26)(63x^2-98x+30)$
Solutions
Question 1
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b)
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l)
Question 2
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Question 3
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Question 4
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Question 5
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Question 6
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Question 7
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Question 8
a) $\begin{align}((x+2)+3)((x+2)-3)&=(x+2)^2-9\\&=x^2+4x-5\end{align}$
b) $\begin{align}((x+1)+5)((x+1)-5)&=(x+1)^2-25\\&=x^2+2x-24\end{align}$
c) $\begin{align}((x+3)+8)((x+2)-8)&=(x+3)^2-64\\&=x^2+6x-55\end{align}$
d)
e)
f)
Question 9
a) $\begin{align}x^4-1&=(x^2-1)(x^2+1)\\&=(x-1)(x+1)(x^2+1)\end{align}$
b) $\begin{align}16x^4-1&=(4x^2-1)(4x^2+1)\\&=(2x-1)(2x+1)(4x^2+1)\end{align}$
c) $\begin{align}x^8-256&=(x^4-16)(x^4+16)\\&=(x^2-4)(x^2+4)(x^4+16)\\&=(x-2)(x+2)(x^2+4)(x^4+16)\end{align}$
d) $\begin{align}243x^4-48&=3(81x^4-16)\\&=3(9x^2-4)(9x^2+4)\\&=3(3x-2)(3x+2)(9x^2+4)\end{align}$
Question 10
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b)
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Question 11
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b)
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g) $\begin{align}25(x+5)^2-100(2x-3)^2&=(5(x+5)+10(2x-3))(5(x+5)-10(2x-3))\\&=(5x+25+20x-30)(5x+25-20x+30)\\&=(25x-5)(55-15x)\\&=5(5x-1)5(11-3x)\\&=25(5x-1)(11-3x)\end{align}$
h) $\begin{align}49(3x-2)^4-4(7x-1)^2&=(7(3x-2)^2+2(7x-1))(7(3x-2)^2-2(7x-1))\\&=(7(9x^2-12x+4)+14x-2)(7(9x^2-12x+4)-14x+2)\\&=(63x^2-84x+28+14x-2)(63x^2-84x+28-14x+2)\\&=(63x^2-70x+26)(63x^2-98x+30)\end{align}$