Factorise a Polynomial (HCF)

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Question 1 of 7

1. Question
Factorise.

$40 x^{2} + 35 x$ $40x^2+35x$
- 1. $10 x (3 x + 4)$ $10x(3x+4)$
- 2. $5 x (2 x + 3 x)$ $5x(2x+3x)$
- 3. $4 x (5 x + 6)$ $4x(5x+6)$
- 4. $5 x (8 x + 7)$ $5x(8x+7)$
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A Greatest Common Factor is the factor of two terms with the highest value.

First, find the Greatest Common Factor ( $G C F$ $GCF$ ) of the two terms by using a factor tree.

Factors of $40 x^{2}$ $40x^2$ :

Factors of $35 x$ $35x$ :

Collect the common factors and multiply them all to get the GCF.

GCF $=$ $=$ $5$ $5$ $\times$ $times$ $x$ $x$

$=$ $=$ $5 x$ $5x$

Finally, factorise by placing $5 x$ $5x$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by $5 x$ $5x$ , then simplify.

$5 x [(40 x^{2} \div 5 x) + (35 x \div 5 x)]$ $5x[(40x^2div5x)+(35xdiv5x)]$

$=$ $=$ $5 x (8 x + 7)$ $5x(8x+7)$

$5 x (8 x + 7)$ $5x(8x+7)$
Question 2 of 7

2. Question
Factorise.

$28 x^{2} y^{2} - 49 x y z$ $28x^2y^2-49xyz$
- 1. $7 x y (4 x y - 7 z)$ $7xy(4xy-7z)$
- 2. $7 y (14 x - 21 x z)$ $7y(14x-21xz)$
- 3. $7 x y z (4 x y - 7)$ $7xyz(4xy-7)$
- 4. $7 x (21 x + 42 z)$ $7x(21x+42z)$
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A Greatest Common Factor is the factor of two terms with the highest value.

First, find the Greatest Common Factor ( $G C F$ $GCF$ ) of the two terms by using a factor tree.

Factors of $28 x^{2} y^{2}$ $28x^2y^2$ :

Factors of $49 x y z$ $49xyz$ :

Collect the common factors and multiply them all to get the GCF.

GCF $=$ $=$ $7$ $7$ $\times$ $times$ $x$ $x$ $\times$ $times$ $y$ $y$

$=$ $=$ $7 x y$ $7xy$

Finally, factorise by placing $7 x y$ $7xy$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by $7 x y$ $7xy$ , then simplify.

$7 x y [(28 x^{2} y^{2} \div 7 x y) - (49 x y z \div 7 x y)]$ $7xy[(28x^2y^2div7xy)-(49xyzdiv7xy)]$

$=$ $=$ $7 x y (4 x y - 7 z)$ $7xy(4xy-7z)$

$7 x y (4 x y - 7 z)$ $7xy(4xy-7z)$
Question 3 of 7

3. Question
Factorise.

$8 m p - 16 m t + 2 m^{2}$ $8mp-16mt+2m^2$
- 1. $2 m (4 p - 8 t + m)$ $2m(4p-8t+m)$
- 2. $4 m (2 p - 4 t + m)$ $4m(2p-4t+m)$
- 3. $4 m (p - 2 t + m)$ $4m(p-2t+m)$
- 4. $2 m (6 p - 10 t + 2 m)$ $2m(6p-10t+2m)$
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A Greatest Common Factor is the factor of two terms with the highest value.

First, find the Greatest Common Factor ( $G C F$ $GCF$ ) of the three terms by using a factor tree.

Factors of $8 m p$ $8mp$ :

Factors of $16 m t$ $16mt$ :

Factors of $2 m^{2}$ $2m^2$ :

Collect the common factors and multiply them all to get the GCF.

GCF $=$ $=$ $2$ $2$ $\times$ $times$ $m$ $m$

$=$ $=$ $2 m$ $2m$

Finally, factorise by placing $2 m$ $2m$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by $2 m$ $2m$ , then simplify.

$2 m [(8 m p \div 2 m) - (16 m t \div 2 m) + (2 m^{2} \div 2 m)]$ $2m[(8mpdiv2m)-(16mtdiv2m)+(2m^2div2m)]$

$=$ $=$ $2 m (4 p - 8 t + m)$ $2m(4p-8t+m)$

$2 m (4 p - 8 t + m)$ $2m(4p-8t+m)$
Question 4 of 7

4. Question
Factorise.

$36 a^{2} b - 54 a b^{2} c$ $36a^2b-54ab^2c$
- 1. $12 a b (2 a^{2} - 3 c)$ $12ab(2a^2-3c)$
- 2. $18 a b (2 a b - 3 a c)$ $18ab(2ab-3ac)$
- 3. $12 a b (3 a - 4 b c)$ $12ab(3a-4bc)$
- 4. $18 a b (2 a - 3 b c)$ $18ab(2a-3bc)$
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A Greatest Common Factor is the factor of two terms with the highest value.

First, find the Greatest Common Factor ( $G C F$ $GCF$ ) of the two terms by using a factor tree.

Factors of $36 a^{2} b$ $36a^2b$ :

Factors of $54 a b^{2} c$ $54ab^2c$ :

Collect the common factors and multiply them all to get the GCF.

GCF $=$ $=$ $2$ $2$ $\times$ $times$ $3$ $3$ $\times$ $times$ $3$ $3$ $\times$ $times$ $a$ $a$ $\times$ $times$ $b$ $b$

$=$ $=$ $18 a b$ $18ab$

Finally, factorise by placing $18 a b$ $18ab$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by $18 a b$ $18ab$ , then simplify.

$18 a b [(36 a^{2} b \div 18 a b) - (54 a b^{2} c \div 18 a b)]$ $18ab[(36a^2bdiv18ab)-(54ab^2cdiv18ab)]$

$=$ $=$ $18 a b (2 a - 3 b c)$ $18ab(2a-3bc)$

$18 a b (2 a - 3 b c)$ $18ab(2a-3bc)$
Question 5 of 7

5. Question
Factorise.

$9 x y^{2} + 18 x^{2} y + 12 x y$ $9xy^2+18x^2y+12xy$
- 1. $3 x y (18 x y + 15 y + 15 x)$ $3xy(18xy+15y+15x)$
- 2. $6 x y (y + 2 x + 3)$ $6xy(y+2x+3)$
- 3. $12 x y (y + 6 x)$ $12xy(y+6x)$
- 4. $3 x y (3 y + 6 x + 4)$ $3xy(3y+6x+4)$
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A Greatest Common Factor is the factor of two terms with the highest value.

First, find the Greatest Common Factor ( $G C F$ $GCF$ ) of the three terms by using a factor tree.

Factors of $9 x y^{2}$ $9xy^2$ :

Factors of $18 x^{2} y$ $18x^2y$ :

Factors of $12 x y$ $12xy$ :

Collect the common factors and multiply them all to get the GCF.

GCF $=$ $=$ $3$ $3$ $\times$ $times$ $x$ $x$ $\times$ $times$ $y$ $y$

$=$ $=$ $3 x y$ $3xy$

Finally, factorise by placing $3 x y$ $3xy$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by $3 x y$ $3xy$ , then simplify.

$3 x y [(9 x y^{2} \div 3 x y) + (18 x^{2} y \div 3 x y) + (12 x y \div 3 x y)]$ $3xy[(9xy^2div3xy)+(18x^2ydiv3xy)+(12xydiv3xy)]$

$=$ $=$ $3 x y (3 y + 6 x + 4)$ $3xy(3y+6x+4)$

$3 x y (3 y + 6 x + 4)$ $3xy(3y+6x+4)$
Question 6 of 7

6. Question
Factorise.

$24 n - 60 m n - 96 n^{2}$ $24n-60mn-96n^2$
- 1. $12 n (2 - 5 m - 8 n)$ $12n(2-5m-8n)$
- 2. $6 n (4 - 10 m - 16 n)$ $6n(4-10m-16n)$
- 3. $8 n (3 - 7 m - 12 n)$ $8n(3-7m-12n)$
- 4. $9 n (3 - 10 m - 12 n)$ $9n(3-10m-12n)$
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A Greatest Common Factor is the factor of two terms with the highest value.

First, find the Greatest Common Factor ( $G C F$ $GCF$ ) of the three terms by using a factor tree.

Factors of $24 n$ $24n$ :

Factors of $60 m n$ $60mn$ :

Factors of $96 n^{2}$ $96n^2$ :

Collect the common factors and multiply them all to get the GCF.

GCF $=$ $=$ $3$ $3$ $\times$ $times$ $2$ $2$ $\times$ $times$ $2$ $2$ $\times$ $times$ $n$ $n$

$=$ $=$ $12 n$ $12n$

Finally, factorise by placing $12 n$ $12n$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by $12 n$ $12n$ , then simplify.

$12 n [(24 n \div 12 n) - (60 m n \div 12 n) - (96 n^{2} \div 12 n)]$ $12n[(24ndiv12n)-(60mndiv12n)-(96n^2div12n)]$

$=$ $=$ $12 n (2 - 5 m - 8 n)$ $12n(2-5m-8n)$

$12 n (2 - 5 m - 8 n)$ $12n(2-5m-8n)$
Question 7 of 7

7. Question
Factorise.

$16 p q^{2} + 20 p^{2} q v - 24 q^{2}$ $16pq^2+20p^2qv-24q^2$
- 1. $2 q (5 p q - 10 p^{2} - 6 q)$ $2q(5pq-10p^2-6q)$
- 2. $4 q (8 q + 2 p q^{2} - 3 q)$ $4q(8q+2pq^2-3q)$
- 3. $4 q (4 p q + 5 p^{2} v - 6 q)$ $4q(4pq+5p^2v-6q)$
- 4. $4 q (4 p q + 5 p v^{2} + 10 q)$ $4q(4pq+5pv^2+10q)$
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A Greatest Common Factor is the factor of two terms with the highest value.

First, find the Greatest Common Factor ( $G C F$ $GCF$ ) of the three terms by using a factor tree.

Factors of $16 p q^{2}$ $16pq^2$ :

Factors of $20 p^{2} q v$ $20p^2qv$ :

Factors of $24 q^{2}$ $24q^2$ :

Collect the common factors and multiply them all to get the GCF.

GCF $=$ $=$ $2$ $2$ $\times$ $times$ $2$ $2$ $\times$ $times$ $q$ $q$

$=$ $=$ $4 q$ $4q$

Finally, factorise by placing $4 q$ $4q$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by $4 q$ $4q$ , then simplify.

$4 q [(16 p q^{2} \div 4 q) + (20 p^{2} q v \div 4 q) - (24 q^{2} \div 4 q)]$ $4q[(16pq^2div4q)+(20p^2qvdiv4q)-(24q^2div4q)]$

$=$ $=$ $4 q (4 p q + 5 p^{2} v - 6 q)$ $4q(4pq+5p^2v-6q)$

$4 q (4 p q + 5 p^{2} v - 6 q)$ $4q(4pq+5p^2v-6q)$

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Quizzes

GCF	$=$ $=$	$5$ $5$ $\times$ $times$ $x$ $x$
	$=$ $=$	$5 x$ $5x$

		$5 x [(40 x^{2} \div 5 x) + (35 x \div 5 x)]$ $5x[(40x^2div5x)+(35xdiv5x)]$
	$=$ $=$	$5 x (8 x + 7)$ $5x(8x+7)$

GCF	$=$ $=$	$7$ $7$ $\times$ $times$ $x$ $x$ $\times$ $times$ $y$ $y$
	$=$ $=$	$7 x y$ $7xy$

		$7 x y [(28 x^{2} y^{2} \div 7 x y) - (49 x y z \div 7 x y)]$ $7xy[(28x^2y^2div7xy)-(49xyzdiv7xy)]$
	$=$ $=$	$7 x y (4 x y - 7 z)$ $7xy(4xy-7z)$

GCF	$=$ $=$	$2$ $2$ $\times$ $times$ $m$ $m$
	$=$ $=$	$2 m$ $2m$

		$2 m [(8 m p \div 2 m) - (16 m t \div 2 m) + (2 m^{2} \div 2 m)]$ $2m[(8mpdiv2m)-(16mtdiv2m)+(2m^2div2m)]$
	$=$ $=$	$2 m (4 p - 8 t + m)$ $2m(4p-8t+m)$

GCF	$=$ $=$	$2$ $2$ $\times$ $times$ $3$ $3$ $\times$ $times$ $3$ $3$ $\times$ $times$ $a$ $a$ $\times$ $times$ $b$ $b$
	$=$ $=$	$18 a b$ $18ab$

		$18 a b [(36 a^{2} b \div 18 a b) - (54 a b^{2} c \div 18 a b)]$ $18ab[(36a^2bdiv18ab)-(54ab^2cdiv18ab)]$
	$=$ $=$	$18 a b (2 a - 3 b c)$ $18ab(2a-3bc)$

GCF	$=$ $=$	$3$ $3$ $\times$ $times$ $x$ $x$ $\times$ $times$ $y$ $y$
	$=$ $=$	$3 x y$ $3xy$

		$3 x y [(9 x y^{2} \div 3 x y) + (18 x^{2} y \div 3 x y) + (12 x y \div 3 x y)]$ $3xy[(9xy^2div3xy)+(18x^2ydiv3xy)+(12xydiv3xy)]$
	$=$ $=$	$3 x y (3 y + 6 x + 4)$ $3xy(3y+6x+4)$

		$12 n [(24 n \div 12 n) - (60 m n \div 12 n) - (96 n^{2} \div 12 n)]$ $12n[(24ndiv12n)-(60mndiv12n)-(96n^2div12n)]$
	$=$ $=$	$12 n (2 - 5 m - 8 n)$ $12n(2-5m-8n)$

		$4 q [(16 p q^{2} \div 4 q) + (20 p^{2} q v \div 4 q) - (24 q^{2} \div 4 q)]$ $4q[(16pq^2div4q)+(20p^2qvdiv4q)-(24q^2div4q)]$
	$=$ $=$	$4 q (4 p q + 5 p^{2} v - 6 q)$ $4q(4pq+5p^2v-6q)$

GCF	$=$ $=$	$3$ $3$ $\times$ $times$ $2$ $2$ $\times$ $times$ $2$ $2$ $\times$ $times$ $n$ $n$
	$=$ $=$	$12 n$ $12n$

GCF	$=$ $=$	$2$ $2$ $\times$ $times$ $2$ $2$ $\times$ $times$ $q$ $q$
	$=$ $=$	$4 q$ $4q$