Question 12 Marks
Divide as directed: $x (x + 1) (x + 2) (x + 3)$ $\div$ $x (x + 1)$
Answer$x (x + 1) (x + 2) (x + 3)$ $\div$ $x (x + 1)$
$ = \frac{{x(x + 1)(x + 2)(x + 3)}}{{x(x + 1)}}$
$= (x + 2) (x + 3)$
View full question & answer→Question 22 Marks
Divide as directed: $20(y + 4) (y^2+ 5y + 3) \div 5(y + 4)$
Answer$20(y + 4) (y^2+ 5y + 3) \div 5(y + 4)$
$ = \frac{{20(y + 4)({y^2} + 5y + 3)}}{{5(y + 4)}}$
$= 4(y^2+ 5y + 3)$
View full question & answer→Question 32 Marks
Divide as directed: $52pqr\ (p + q) (q + r) (r + p)$ $\div$ $104pq\ (q + r) (r + p)$
AnswerWe have $52pqr (p + q) (q + r)(r + p)$ $\div$ $104pq\ (q + r) (r + p)$
$={52pqr(p+q)(q+r)(r+p) \over 104pq(q+r)(r+p)}$
$ = {r(p+q)\over 2}$
View full question & answer→Question 42 Marks
Divide as directed: $26xy (x + 5) (y – 4 )$ $\div$ $13x (y – 4)$
Answer$26xy (x + 5) (y – 4 )$ $\div$ $13x (y – 4)$
$ = \frac{{26xy(x + 5)(y - 4)}}{{13x(y - 4)}}$
$= 2y(x + 5)$
View full question & answer→Question 52 Marks
Divide as directed: $5(2x + 1) (3x + 5)$ $\div$ $(2x + 1)$
Answer$5(2x + 1) (3x + 5)$ $\div$ $(2x + 1)$
$ = \frac{{5(2x + 1)(3x + 5)}}{{2x + 1}}$
$= 5(3x + 5)$
View full question & answer→Question 62 Marks
Work out the division: $96abc\ (3a – 12) (5b – 30)$ $\div$ $144\ (a – 4) (b – 6)$
Answer$96abc (3a – 12) (5b – 30)$ $\div$ $144 (a – 4) (b – 6)$
$ = \frac{{96abc(3a - 12)(5b - 30)}}{{144(a - b)(b - 6)}}$
$ = \frac{{96abc \times 3(a - 4) \times 5(b - 6)}}{{144(a - 4)(b - 6)}}$
$= 10abc$
View full question & answer→Question 72 Marks
Work out the division: $9x^2y^2(3z – 24)$ $\div$ $27xy (z – 8)$
Answer$9x^2y^2(3z – 24)$ $\div$ $27xy (z – 8)$
$ = \frac{{9{x^2}{y^2}(3z - 24)}}{{27xy(z - 8)}}$
$ = {9x^2 y^2 3(z-8) \over 27xy(z-8)}$
= xy
View full question & answer→Question 82 Marks
Work out the division : $10y(6y + 21)$ $\div$ $5(2y + 7)$
Answer$10y (6y + 21)$ $\div$ $5(2y + 7)$
$ = \frac{{10y(6y + 21)}}{{5(2y + 7)}}$
$ = \frac{{10y \times 3(2y + 7)}}{{5(2y + 7)}}$
$= 6y$
View full question & answer→Question 92 Marks
Work out the division: $(10x – 25)$ $\div$ $(2x – 5)$
Answer$(10x – 25)$ $\div$ $(2x – 5)$
$ = \frac{{10x - 25}}{{2x - 5}}$
$ = \frac{{5(2x - 5)}}{{(2x - 5)}}$
$= 5$
View full question & answer→Question 102 Marks
Work out the division: $(10x – 25) ÷ 5$
Answer$(10x – 25) ÷ 5$
$ = \frac{{10x - 25}}{5}$
$ = \frac{{5(2x - 5)}}{5}$
$= (2x – 5)$
View full question & answer→Question 112 Marks
Divide the given polynomial by the given monomial: $\left(p^3 q^6-p^6 q^3\right) \div p^3 q^3$
Answer$\left(p^3 q^6-p^6 q^3\right) \div p^3 q^3$
$ = \frac{{{p^3}{q^6} - {p^6}{q^3}}}{{{p^3}{q^3}}}$
$ = \frac{{{p^3}{q^3}({q^3} - {p^3})}}{{{p^3}{q^3}}}$
$= q^3– p^3$
View full question & answer→Question 122 Marks
Divide the given polynomial by the given monomial: $\left(x^3+2 x^2+3 x\right) \div 2 x$
Answer$\left(x^3+2 x^2+3 x\right) \div 2 x$
$ = \frac{{{x^3} + 2{x^2} + 3x}}{{2x}}$
$ = \frac{{x({x^2} + 2x + 3)}}{{2x}}$
$ = x^2 + 2x+ 3 \over 2$
View full question & answer→Question 132 Marks
Divide the given polynomial by the given monomial: $8\left(x^3 y^2 z^2+x^2 y^3 z^2+x^2 y^2 z^3\right) \div 4 x^2 y^2 z^2$
Answer$8\left(x^3 y^2 z^2+x^2 y^3 z^2+x^2 y^2 z^3\right) \div 4 x^2 y^2 z^2$
$ = \frac{{8({x^3}{y^2}{z^2} + {x^2}{y^2}{z^2} + {x^2}{y^2}{z^2})}}{{4{x^2}{y^2}{z^2}}}$
$ = \frac{{8{x^2}{y^2}{z^2}(x + y + z)}}{{4{x^2}{y^2}{z^2}}}$
$= 2(x + y + z)$
View full question & answer→Question 142 Marks
Divide the given polynomial by the given monomial: $\left(3 y^8-4 y^6+5 y^4\right) \div y^4$
Answer$\left(3 y^8-4 y^6+5 y^4\right) \div y^4$
$= {3y^8- 4y^6 +5y^4} \over y^4$
$ = \frac{{{y^4}(3{y^4} - 4{y^2} + 5)}}{{{y^4}}}$
$= 3y^4– 4y^2+ 5$
View full question & answer→Question 152 Marks
Divide the given polynomial by the given monomial: $(5x^2– 6x) \div 3x$
Answer$(5x^2– 6x) \div 3x$
$ = \frac{{5{x^2} - 6x}}{{3x}}$
$ = \frac{{x(5x - 6)}}{{3x}}$
$ = \frac{1}{3}(5x - 6)$
View full question & answer→Question 162 Marks
Factorise: $x^4-(y+z)^4$
Answer$x^4-(y+z)^4$
$\left(x^2\right)^2-\left\{(y+z)^2\right\}^2. . . .$ [Using Identity $III$
$\left\{x^2-(y+z)^2\right\}\left\{x^2+(y+z)^2\right\}. . . .$[Using Identity $III$
$= (x-y-z)(x+y+z)\left\{x^2+(y+z)^2\right\}$
View full question & answer→Question 172 Marks
Factorise: $p^4– 81$
Answer$p^4-81$
$=\left(p^2\right)^2-(9)^2$
$\left(p^2-9\right)\left(p^2+9\right) . . . .$[Using Identity $III$
$ =\left\{(p)^2-(3)^2\right\}\left(p^2+9\right)$
$=(p-3)(p+3)\left(p^2+9\right). . . .$[Using Identity $III$
View full question & answer→Question 182 Marks
Factorise: $a^4– b^4$
Answer$a^4– b^4$
$= (a^2)^2– (b^2)^2$
$= (a^2– b^2) (a^2+ b^2)$
$= (a – b) (a + b) (a^2+ b^2). . . . $[Using Identity $III$
View full question & answer→Question 192 Marks
Factorise the expressions: $6xy – 4y + 6 – 9x$
Answer$6xy – 4y + 6 – 9x$
$= 6xy – 4y – 9x + 6$
$= 2y (3x – 2) – 3(3x – 2)$
$= (3x – 2) (2y – 3)$
View full question & answer→Question 202 Marks
Factorise the expressions: $10ab + 4a + 5b + 2$
Answer$10ab + 4a + 5b + 2$
$= 2a (5b + 2) + 1 (5b + 2)$
$= (5b + 2) (2a + 1)$
View full question & answer→Question 212 Marks
Factorise the expressions: $5y^2– 20y – 8z + 2yz$
Answer$5y^2– 20y – 8z + 2yz$
$= 5y^2– 20y + 2yz – 8z$
$= (y – 4) (5y + 2z)$
View full question & answer→Question 222 Marks
Factorise the expressions: $y (y + z) + 9 (y + z)$
Answer$y (y + z) + 9 (y + z)$
$= (y + z) (y + 9)$
View full question & answer→Question 232 Marks
Factorise the expressions: $(1m + 1) + m + 1$
Answer$(1m + 1) + m + 1$
$= 1 (m + 1) + 1(m + 1)$
$= (m + 1) (1 + 1)$
$=2(m+1)$
View full question & answer→Question 242 Marks
Factorise the expressions: $ a m^2+b m^2+b n^2+a n^2 $
Answer$ a m^2+b m^2+b n^2+a n^2 $
$ a m^2+b m^2+b n^2+a n^2=a m^2+b m^2+a n^2+b n^2 $
$ =m^2(a+b)+n^2(a+b) $
$ =(a+b)\left(m^2+n^2\right) $
View full question & answer→Question 252 Marks
Factorise the expressions: $ 2 x^3+2 x y^2+2 x z^2 $
Answer$ 2 x^3+2 x y^2+2 x z^2 $
$ 2 x^3+2 x y^2+2 x z^2=2 x\left(x^2+y^2+z^2\right) $
View full question & answer→Question 262 Marks
Factorise the expressions: $ 7 p^2+21 q^2 $
Answer$ 7 p^2+21 q^2 $
$ 7 p^2+21 q^2=7\left(p^2+3 q^2\right) $
View full question & answer→Question 272 Marks
Factorise the expressions: $ax^2+ bx$
Answer$ax^2+ bx$
$ax^2+ bx= x(ax + b)$
View full question & answer→Question 282 Marks
Factorise: $(x^2– 2xy +y^2) – z^2$
Answer$(x^2– 2xy +y^2) – z^2$
$= (x – y)^2– z^2. . . .$ [Using Identity $II$
$=(x – y – z) (x – y + z). . . .$ [Using Identity $III$
View full question & answer→Question 292 Marks
Factorise: $9x^2y^2– 16$
Answer$9x^2y^2–16$
$= (3xy)^2– (4)^2$
$= (3xy – 4) (3xy + 4). . . . $[Using Identity]
View full question & answer→Question 302 Marks
Factorise: $(l + m)^2– (l – m)^2$
Answer$(l + m)^2– (l – m)^2$
$= l{(l + m) – (l – m)} {(l + m) + (l – m)} . . . .$ [Applying Identity $III$
$= (2m) (2l)$
$= 4lm$
View full question & answer→Question 312 Marks
Factorise: $16x^5– 144x^3$
Answer$ 16 x^5-144 x^3 $
$ =16 x^3\left(x^2-9\right) $
$ =16 x^3\left\{(x)^2-(3)^2\right\} $
$ =16 x^3(x-3)(x+3) $
View full question & answer→Question 322 Marks
Factorise: $49x^2– 36$
Answer$49x^2– 36$
$= (7x)^2– (6)^2$
$= (7x – 6) (7x + 6). . . . $[Using Identity $III$
View full question & answer→Question 332 Marks
Factorise: $63a^2– 112b^2$
Answer$63a^2– 112b^2$
$= 7(9a^2– 16b^2)$
$= 7 {(3a)^2– (4b)^2}$
$= 7 (3a – 4b) (3a + 4b). . . . [$Applying Identity $III]$
View full question & answer→Question 342 Marks
Factorise: $4p^2– 9q^2$
Answer$4p^2– 9q^2$
$= (2p)^2– (3q)^2$
$= (2p – 3q) (2p + 3q). . . . [$Using Identity $III$
View full question & answer→Question 352 Marks
Factorise the expression: $a^4+ 2a^2b^2+ b^4$
Answer$a^4+ 2a^2b^2+ b^4$
$= (a^2)^2+ 2(a^2) (b^2) + (b^2)^2$
$= (a^2+ b^2)^2 . . . [$Using Identity $I$
View full question & answer→Question 362 Marks
Factorise the expression $121b^2– 88bc + 16c^2$
Answer$121b^2– 88bc + 16c^2$
$= (11b)^2– 2(11b) (4c) + (4c)^2$
$= (11b – 4c)^2. . . . [$Using Identity $II]$
View full question & answer→Question 372 Marks
Factorise the expression : $4x^2– 8x + 4$
Answer$4x^2– 8x + 4$
$= 4(x^2– 2x + 1)$
$= 4[(x)^2– 2(x) (1) + (1)^2]$
$= 4(x – 1)^2. . . . [$Applying Identity $II]$
View full question & answer→Question 382 Marks
Factorise the expression :$49y^2+ 84yz + 36z^2$
Answer$49y^2+ 84yz + 36z^2$
$= (7y)^2+ 2(7y) (6z) + (6z)^2$
$= (7y + 6z)^2. . . .[$ Using Identity $I]$
View full question & answer→Question 392 Marks
Factorise the expression : $25m^2+ 30m + 9$
Answer$25m^2+ 30m + 9$
$= (5m)^2+ 2(5m) (3) + (3)^2$
$= (5m + 3)^2.... [$Applying Identity I]$
View full question & answer→Question 402 Marks
Factorise the expression: $p^2– 10p + 25$
Answer$p^2-10 p+25=p^2+(-5-5) p+(-5)(-5)$
Using identity $x^2+(a+b) x+a b=(x+a)(x+b)$,
Here $x = p, a = -5$ and $b = -5$
So, $p^2-10 p+25=(p-5)(p-5)$
View full question & answer→Question 412 Marks
Factorise the expression: $a^2+ 8a + 16$
Answer$a^2+ 8a + 16$
$= (a^2) + 2(4) (a) + (4)^2$
$= (a + 4)^2. . . . [$ Applying Identity $I]$
View full question & answer→Question 422 Marks
Factorise :$z – 7 + 7xy – xyz$
Answer$z – 7 + 7xy – xyz$
$= z – 7 – xyz + 7xy$
$= 1(z – 7) – xy (z – 7)$
$= (z – 7) (1 – xy).$
View full question & answer→Question 432 Marks
Factorise : $15pq + 15 + 9q + 25p$
Answer$15pq + 15 + 9q + 25p$
$= 15pq + 9q + 25p + 15$
$= 3q (5p + 3) + 5(5p + 3)$
$= (5p + 3) (3q + 5)$
View full question & answer→Question 442 Marks
Factorise : $ax + bx – ay – by$
Answer$ax + bx – ay – by$
$= x (a + b) – y (a + b)$
$= (a + b)(x – y)$
View full question & answer→Question 452 Marks
Factorise :$15xy – 6x + 5y – 2$
Answer$15xy – 6x + 5y – 2$
$= 3x (5y – 2) + 1 (5y – 2)$
$= (5y – 2) (3x + 1)$
View full question & answer→Question 462 Marks
Factorise :$x^2+ xy + 8x + 8y$
Answer$x^2+ xy + 8x + 8y$
$= x (x + y) + 8 (x + y)$
$= (x + y) (x + 8)$
View full question & answer→Question 472 Marks
Factorise the expression: $x^2yz + xy^2z + xyz^2$
Answer$x^2yz + xyz + xyz^2= x\times x \times y \times z + x \times y \times y \times z + x \times y \times z \times z$
Taking common factors from each term,
$= x \times y \times z(x + y + z)$
$= xyz(x + y + z)$
View full question & answer→Question 482 Marks
Factorise the expression: $10a^2- 15b^2+ 20c^2$
Answer$10a^2- 15b^2+ 20c^2= 2 \times 5 \times a \times a - 3 \times 5 \times b \times b + 2 \times 2 \times 5 \times c \times c$
Taking common factors from each term,
$= 5(2 \times a \times a - 3 \times b \times b + 2 \times 2 \times c \times c)$
$= 5(2a^2- 3b^2+ 4c^2)$
View full question & answer→Question 492 Marks
Factorise the expression: $20l^2m + 30alm$
Answer$20l^2m + 30alm = 2 \times 2 \times 5 \times l \times l \times m + 2 \times 3 \times 5 \times a \times l \times m$
Taking common factors from each term,
$= 2 \times 5 \times l \times m(2 \times l + 3 \times a)$
$= 10lm(2l + 3a)$
View full question & answer→Question 502 Marks
Factorise the expression: $ax^2y + bxy^2+ cxyz$
Answer$ax^2y + bxy^2+ cxyz = a$
$ \times x \times x \times y + b \times x \times y \times y + c \times x \times y \times z$
Taking common factors from each term,
$= x \times y(a \times x + b \times y + c \times z)$
$= xy(ax + by + cz)$
View full question & answer→Question 512 Marks
Find the common factors of the given term: $3 x^2 y^3, 10 x^3 y^2, 6 x^2 y^2 z$
Answer$3 x^2 y^3=$ $3\times x\times x\times y\times y\times y$
$10x^3y^2=$ $2\times 5\times x\times x\times x\times y\times y$
$6x^2y^2z =$ $2\times 3\times x\times x\times y\times y\times z$
Hence the common factors are $x, x, y, y$
and $x\times x\times y\times y$ $= x^2y^2$
View full question & answer→Question 522 Marks
Find the common factors of the given term: $10pq, 20qr, 30rp$
Answer$10pq$ $ = \underline 2 \times \underline{\underline 5} \times p \times q$
$20qr$ $ = \underline 2 \times 2 \times \underline{\underline 5} \times q \times r$
$30rp$ $ = \underline 2 \times 3 \times \underline{\underline 5} \times r \times p$
Common prime factors are $2$ and $5$
$\therefore H.C.F. = 2 \times 5 = 10$
View full question & answer→Question 532 Marks
Find the common factors of the given term: $16 x^{3},-4 x^{2}, 32 x$
AnswerThe given terms $16 x^{3},-4 x^{2}, 32 x$ can be written as:
$16x^3=$ $2 \times 2 \times 2 \times 2 \times x \times x \times x$
$-4x^2=$ $-1 \times 2 \times 2 \times x \times x$
$32x =$ $2 \times 2 \times 2 \times 2 \times 2 \times x$
The common factors are $2, 2$ and $x$ = $2 \times 2 \times x$ = 4x
View full question & answer→Question 542 Marks
Find the common factors of the given term: $6abc, 24ab^2, 12a^2b$
Answer6abc $= 2 × 3 × a × b × c$
$24ab^2 = 2 × 2 × 2 × 3 × a × b × b$
$12a^2b = 2 × 2 × 3 × a × a × b$
Common prime factors are $2, 3, a$ and $b$
$\therefore H.C.F. = 2 \times 3 \times a \times b$
$= 6ab$
View full question & answer→Question 552 Marks
Find the common factors of the given term: $2x, 3x^2, 4$
Answer$2x, 3x^2, 4$
$2x = \underline 1 \times 2 \times \underline{\underline x} $
$3{x^2} = \underline 1 \times 3 \times \underline{\underline x} \times x$
$4 = 1 \times 2 \times 2$
Common factors are $1$ and $x$.
$\therefore H.C.F. = 1 \times x = x$
View full question & answer→Question 562 Marks
Find the common factors of the given term: $14pq, 28p^2q^2$
Answer$14pq, 28p^2q^2$
14pq $ = \underline 2 \times \underline{\underline 7} \times \mathop p\limits_o \times \mathop q\limits_w $
$28p^2q^2$ $ = \underline 2 \times 2 \times \underline{\underline 7} \times \mathop p\limits_o \times p \times \mathop q\limits_w \times q$
Common prime factors are $2, 7, p$ and $q$.
$\therefore H.C.F. = 2 \times 7 \times p \times q = 14pq$
View full question & answer→Question 572 Marks
Find the common factors of the given term: $2y, 22xy$
Answer$2y, 22xy$
$2y = 2 \times y$
$22 = 2 \times 11 \times x \times y$
Common prime factors are $2$ and $y$.
$\therefore H.C.F. = 2 \times y = 2y$
View full question & answer→Question 582 Marks
Find the common factors of the given term: $12x, 36$
Answer$12x, 36$
$12x = 2 \times 2 \times 3 \times x$
$36 = 2 \times 2 \times 3 \times 3$
Common prime factors are $2$ (occurs twice) and $3$.
$\therefore H.C.F. = 2 \times 2 \times 3 = 12$
View full question & answer→Question 592 Marks
Factorise the expression: $x^2+ 5x + 6$
AnswerWe have $x^2+ 5x + 6,$
$= x^2+ 3x + 2x + 6 = x(x + 3) + 2(x + 3)$
$= (x + 3)(x + 2)$
View full question & answer→Question 602 Marks
Factorise: $a^2– 2ab + b^2– c^2$
Answer$a^2– 2ab + b^2– c^2= (a – b)^2– c^2[$ Using identity, $(x – y)^2= x^2– 2xy + y^2]$
$= {(a – b) – c)}{(a – b) + c}] [$Using identity, $x^2- y^2= (x - y)(x + y)]$
$= (a – b – c) (a – b + c)$
This is the required factorisation.
View full question & answer→Question 612 Marks
Factorise: $49p^2- 36$
Answer$49p^2– 36 = (7p)^2- (6)^2$
Now using identity, $a^2- b^2= (a - b)(a + b),$
$= (7p - 6)(7p + 6)$
This is the required factorisation.
View full question & answer→Question 622 Marks
Factorise: $4y^2– 12y + 9$, using the identity $a^2- 2ab + b^2= (a - b)^2$
AnswerUsing identity, $a^2-2 a b+b^2=(a-b)^2$
$4 y^2-12 y+9$
$=(2 y)^2-2 \times 3 \times(2 y)+(3)^2$
$=(2 y-3)^2$
= (2y – 3)(2y – 3)
This is the required factorisation.
View full question & answer→Question 632 Marks
Factorise: $x^2+8 x+16$
AnswerUsing identity, $a^2+2 a b+b^2=(a+b)^2$
$x^2+8 x+16$
$=x^2+2(x)(4)+4^2 $
$ =(x+4)^2$
= (x + 4)(x + 4)
This is the required factorisation.
View full question & answer→Question 642 Marks
Factorise: $6xy - 4y + 6 - 9x$
Answer$6xy – 4y + 6 - 9x$
$= 6xy - 4y - 9x + 6$
$= 2y(3x - 2) - 3(3x - 2)$
$= (3x - 2)(2y - 3)$
View full question & answer→Question 652 Marks
Find the division:$ 7x^2y^2z^2\div 14xyz$
Answer$7x^2y^2z^2\div 14xyz$ = $\frac {7\times x\times x\times y\times y\times z\times z}{2\times 7\times x\times y\times z}$
= $\frac{x\times y\times z} {2}$= $\frac {1}{2} xyz$
View full question & answer→Question 662 Marks
Solve the division: $-20x^4$ $\div$ $10x62$
AnswerNow, $-20 \mathrm{x}^4=-2 \times 2 \times 5 \times \mathrm{x} \times \mathrm{x} \times \mathrm{x} \times \mathrm{x}$
and $10 \mathrm{x}^2=2 \times 5 \times \mathrm{x} \times \mathrm{x}$
Therefore, $(–20x^4)$ $\div$ $10x^2=$ $\frac{-2\times2\times5\times x\times x\times x\times x}{2\times5 \times x\times x}$
$ = -2$ $\times$ x $\times$ $x = -2x^2$
View full question & answer→Question 672 Marks
Obtain the factors of $z^2- 4z – 12.$
Answer$z^2– 4z –12 = z^2– 6z + 2z –12$
$= z(z – 6) + 2(z – 6 )$
$= (z – 6)(z + 2)$
Thus, the factors of $z^2– 4z – 12$ are $(z – 6)$ and $(z + 2)$
View full question & answer→Question 682 Marks
Find the factors of $y^2- 7y + 12.$
Answer$y^2– 7y + 12 = y^2– 3y – 4y + 12$
$= y(y –3) – 4(y – 3) = (y – 3)(y – 4)$
Thus, the factors are $(y – 3)$ and $(y – 4)$.
View full question & answer→