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#### Question 1:

Divide 5x3 − 15x2 + 25x by 5x. #### Question 2:

Divide 4z3 + 6z2z by − $\frac{1}{2}$z. #### Question 3:

Divide 9x2y − 6xy + 12xy2 by −$\frac{3}{2}$xy. #### Question 4:

Divide 3x3y2 + 2x2y + 15xy by 3xy. #### Question 5:

Divide x2 + 7x + 12 by x + 4. #### Question 6:

Divide 4y2 + 3y + $\frac{1}{2}$ by 2y + 1. #### Question 7:

Divide 3x3 + 4x2 + 5x + 18 by x + 2. #### Question 8:

Divide 14x2 − 53x + 45 by 7x − 9. #### Question 9:

Divide −21 + 71x − 31x2 − 24x3 by 3 − 8x. #### Question 10:

Divide 3y4 − 3y3 − 4y2 − 4y by y2 − 2y. #### Question 11:

Divide 2y5 + 10y4 + 6y3 + y2 + 5y + 3 by 2y3 + 1. #### Question 12:

Divide x4 − 2x3 + 2x2 + x + 4 by x2 + x + 1. #### Question 13:

Divide m3 − 14m2 + 37m − 26 by m2 − 12m +13. #### Question 14:

Divide x4 + x2 + 1 by x2 + x + 1. #### Question 15:

Divide x5 + x4 + x3 + x2 + x + 1 by x3 + 1. #### Question 16:

Divide 14x3 − 5x2 + 9x − 1 by 2x − 1 and find the quotient and remainder #### Question 17:

Divide 6x3x2 − 10x − 3 by 2x − 3 and find the quotient and remainder. #### Question 18:

Divide 6x3 + 11x2 − 39x − 65 by 3x2 + 13x + 13 and find the quotient and remainder. #### Question 19:

Divide 30x4 + 11x3 − 82x2 − 12x + 48 by 3x2 + 2x − 4 and find the quotient and remainder. #### Question 20:

Divide 9x4 − 4x2 + 4 by 3x2 − 4x + 2 and find the quotient and remainder. $\therefore$ Quotient = 3x2 4x 2 and remainder = 0.

#### Question 21:

Verify the division algorithm i.e. Dividend = Divisor × Quotient + Remainder, in each of the following. Also, write the quotient and remainder.

 Dividend Divisor (i) 14x2 + 13x − 15 7x − 4 (ii) 15z3 − 20z2 + 13z − 12 3z − 6 (iii) 6y5 − 28y3 + 3y2 + 30y − 9 2y2 − 6 (iv) 34x − 22x3 − 12x4 − 10x2 − 75 3x + 7 (v) 15y4 − 16y3 + 9y2 − $\frac{10}{3}$y + 6 3y − 2 (vi) 4y3 + 8y + 8y2 + 7 2y2 − y + 1 (vii) 6y5 + 4y4 + 4y3 + 7y2 + 27y + 6 2y3 + 1

(i) Quotient = 2x + 3
Remainder = $-$3
Divisor = 7x $-$ 4
Divisor $×$ Quotient + Remainder = (7x $-$ 4) (2x + 3) $-$ ​3
= 14x+ 21$-$ 8$-$ 12 $-$ ​3
= 14x2 + 13x $-$ 15
= Dividend
Thus,
Divisor $×$ Quotient + Remainder = Dividend
Hence verified.

(ii) Hence verified.

(iii) Quotient = $3{y}^{3}-5y+\frac{3}{2}$
Remainder = 0
Divisor = 2y2 $-$ 6
Divisor $×$ Quotient + Remainder =

= Dividend

Thus, Divisor $×$ Quotient + Remainder = Dividend
Hence verified.

(iv) Quotient  = $-$ 4x3 + 2x2 $-$ 8x + 30
Remainder  = $-$ 285
Divisor  = 3x + 7
Divisor $×$ Quotient + Remainder =  (3x + 7) ($-$ 4x3 + 2x2 $-$ 8x + 30) $-$ 285
= $-$ 12x4 + 6x3 $-$ 24x2 + 90$-$ 28x3 + 14x2 $-$ 56x + 210 $-$ ​285
= $-$ 12x 4 $-$ 22x3 $-$ 10x2 + 34x $-$ 75
=  Dividend
Thus,
Divisor $×$ Quotient + Remainder = Dividend
Hence verified.

(v) Quotient =  $5{y}^{3}-2{y}^{2}+\frac{5}{3}y$
Remainder =  6
Divisor = 3y $-$ 2
Divisor $×$ Quotient  + Remainder = (3y $-$ 2) (5y3 $-$ 2y2 $\frac{5}{3}y$) + 6
= $15{y}^{4}-6{y}^{3}+5{y}^{2}-10{y}^{3}+4{y}^{2}-\frac{10}{3}y+6$
= $15{y}^{4}-16{y}^{3}+9{y}^{2}-\frac{10}{3}y+6$
=  Dividend
Thus,
Divisor $×$ Quotient + Remainder = Dividend
Hence verified.

(vi) Quotient =  2y + 5
Remainder =  11y + 2
Divisor =  2y2 $-$ y + 1
Divisor $×$ Quotient + Remainder =  (2y2 $-$ y + 1) (2y + 5)11y + 2
=  4y3 +10y2 $-$ 2y2 $-$ 5y + 2y + 5 + 11y + 2
=  4y3 + 8y2 + 8y + 7
=  Dividend
Thus,
Divisor $×$ Quotient + Remainder  = Dividend
Hence verified.

(vii) Quotient = 3y2 + 2y + 2
Remainder = 4y2 + 25y + 4
Divisor = 2y3 + 1
Divisor $×$ Quotient + Remainder = (2y3 + 1) (3y2 2y + 2)4y225y + 4
= 6y54y44y33y22y + 4y225y + 4
6y54y44y37y227y + 6
= Dividend
Thus,
Divisor $×$ Quotient + Remainder = Dividend
Hence verified.

#### Question 22:

Divide 15y4 + 16y3 + $\frac{10}{3}$y − 9y2 − 6 by 3y − 2. Write down the coefficients of the terms in the quotient. $\therefore$ Quotient =
5y3 + (26/3)y2 + (25/9)y + (80/27)
Remainder = ($-$ 2/27)
Coefficient of y3 = 5
Coefficient
of y2 = (26/3)
Coefficient of y = (25/9)
Constant = (80/27)

#### Question 23:

Using division of polynomials, state whether
(i) x + 6 is a factor of  x2x − 42
(ii) 4x − 1 is a factor of 4x2 − 13x − 12
(iii) 2y − 5 is a factor of 4y4 − 10y3 − 10y2 + 30y − 15
(iv) 3y2 + 5 is a factor of 6y5 + 15y4 + 16y3 + 4y2 + 10y − 35
(v) z2 + 3 is a factor of z5 − 9z
(vi) 2x2x + 3 is a factor of 6x5x4 + 4x3 − 5x2x − 15

(i) Remainder is zero. Hence (x+6) is a factor of x2 -x-42
(ii) As the remainder is non zero . Hence ( 4x-1) is not a factor of 4x2 -13x-12

(iii) $\because$ The remainder is non zero,
2y $-$ 5 is not a factor of $4{y}^{4}-10{y}^{3}-10{y}^{2}+30y-15$.

(iv) Remainder is zero.  Therefore, 3y2 + 5 is a factor of $6{y}^{5}+15{y}^{4}+16{y}^{3}+4{y}^{2}+10y-35$.

(v) Remainder is zero; therefore, z2 + 3 is a factor of .

(vi) Remainder is zero ; therefore, $2{x}^{2}-x+3$ is a factor of .

#### Question 24:

Find the value of a, if x + 2 is a factor of 4x4 + 2x3 − 3x2 + 8x + 5a.

#### Question 25:

What must be added to x4 + 2x3 − 2x2 + x − 1 , so that the resulting polynomial is exactly divisible by x2 + 2x − 3? Thus, ($-$ 2) should be added to (${x}^{4}+2{x}^{3}-2{x}^{2}+x-1$) to make the resulting polynomial exactly divisible by (${x}^{2}+2x-3$).

#### Question 1:

Divide the first polynomial by the second in each of the following. Also, write the quotient and remainder:
(i) 3x2 + 4x + 5, x − 2
(ii) 10x2 − 7x + 8, 5x − 3
(iii) 5y3 − 6y2 + 6y − 1, 5y − 1
(iv) x4x3 + 5x, x − 1
(v) y4 + y2, y2 − 2

#### Question 2:

Find whether the first polynomial is a factor of the second.
(i) x + 1, 2x2 + 5x + 4
(ii) y − 2, 3y3 + 5y2 + 5y + 2
(iii) 4x2 − 5, 4x4 + 7x2 + 15
(iv) 4 − z, 3z2 − 13z + 4
(v) 2a − 3, 10a2 − 9a − 5
(vi) 4y + 1, 8y2 − 2y + 1

#### Question 1:

Divide:
x2 − 5x + 6 by x − 3

#### Question 2:

Divide:
ax2ay2 by ax + ay

Divide:
x4y4 by x2y2

#### Question 4:

Divide:
acx2 + (bc + ad)x + bd by (ax + b)

#### Question 5:

Divide:
(a2 + 2ab + b2) − (a2 + 2ac + c2) by 2a + b + c

$\phantom{\rule{0ex}{0ex}}\frac{\mathit{\left(}{a}^{\mathit{2}}\mathit{+}\mathit{2}ab\mathit{+}{b}^{\mathit{2}}\mathit{\right)}\mathit{-}\mathit{\left(}{a}^{\mathit{2}}\mathit{+}\mathit{2}ac\mathit{+}{c}^{\mathit{2}}\mathit{\right)}}{\mathit{\left(}\mathit{2}a\mathit{+}b\mathit{+}c\mathit{\right)}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{\left(}a\mathit{+}b{\mathit{\right)}}^{\mathit{2}}\mathit{-}\mathit{\left(}a\mathit{+}c{\mathit{\right)}}^{\mathit{2}}}{\mathit{\left(}\mathit{2}a\mathit{+}b\mathit{+}c\mathit{\right)}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{\left(}a\mathit{+}b\mathit{+}a\mathit{+}c\mathit{\right)}\mathit{\left(}a\mathit{+}b\mathit{-}a\mathit{-}c\mathit{\right)}}{\mathit{\left(}\mathit{2}a\mathit{+}b\mathit{+}c\mathit{\right)}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{\left(}\mathit{2}a\mathit{+}b\mathit{+}c\mathit{\right)}\mathit{\left(}b\mathit{-}c\mathit{\right)}}{\mathit{\left(}\mathit{2}a\mathit{+}b\mathit{+}c\mathit{\right)}}\phantom{\rule{0ex}{0ex}}\mathit{=}b\mathit{-}c$

#### Question 6:

Divide:

$\phantom{\rule{0ex}{0ex}}\frac{\frac{\mathit{1}}{\mathit{4}}{x}^{\mathit{2}}\mathit{-}\frac{\mathit{1}}{\mathit{2}}x\mathit{-}\mathit{12}}{\frac{\mathit{1}}{\mathit{2}}x\mathit{-}\mathit{4}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\frac{\mathit{1}}{\mathit{2}}x\mathit{\left(}\frac{\mathit{1}}{\mathit{2}}x\mathit{-}\mathit{4}\mathit{\right)}\mathit{+}\mathit{3}\mathit{\left(}\frac{\mathit{1}}{\mathit{2}}x\mathit{-}\mathit{4}\mathit{\right)}}{\frac{\mathit{1}}{\mathit{2}}x\mathit{-}\mathit{4}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{\left(}\frac{\mathit{1}}{\mathit{2}}x\mathit{-}\mathit{4}\mathit{\right)}\mathit{\left(}\frac{\mathit{1}}{\mathit{2}}x\mathit{+}\mathit{3}\mathit{\right)}}{\mathit{\left(}\frac{\mathit{1}}{\mathit{2}}x\mathit{-}\mathit{4}\mathit{\right)}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{1}}{\mathit{2}}x\mathit{+}\mathit{3}$

#### Question 1:

Write the degree of each of the following polynomials.
(i) 2x2 + 5x2 − 7
(ii) 5x2 − 3x + 2
(iii) 2x + x2 − 8
(iv) $\frac{1}{2}{y}^{7}-12{y}^{6}+48{y}^{5}-10$
(v) 3x3 + 1
(vi) 5
(vii) 20x3 + 12x2y2 − 10y2 + 20

#### Question 2:

Which of the following expressions are not polynomials?
(i) x2 + 2x−2
(ii) $\sqrt{ax}+{x}^{2}-{x}^{3}$
(iii) 3y3$\sqrt{5}y$ + 9
(iv) ax1/2 + ax + 9x2 + 4
(v) 3x−2 + 2x−1 + 4x +5

#### Question 3:

Write each of the following polynomials in the standard form. Also, write their degree.
(i) x2 + 3 + 6x + 5x4
(ii) a2 + 4 + 5a6
(iii) (x3 − 1)(x3 − 4)
(iv) (y3 − 2)(y3 + 11)
(v) $\left({a}^{3}-\frac{3}{8}\right)\left({a}^{3}+\frac{16}{17}\right)$
(vi) $\left(a+\frac{3}{4}\right)\left(a+\frac{4}{3}\right)$

#### Question 1:

Divide 6x3y2z2 by 3x2yz.

#### Question 2:

Divide 15m2n3 by 5m2n2.

$\phantom{\rule{0ex}{0ex}}\frac{15{m}^{2}{n}^{3}}{5{m}^{2}{n}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{15×m×m×n×n×n}{5×m×m×n×n}\phantom{\rule{0ex}{0ex}}=3{m}^{\left(2-2\right)}{n}^{\left(3-2\right)}\phantom{\rule{0ex}{0ex}}=3{m}^{0}{n}^{1}\phantom{\rule{0ex}{0ex}}=3n$

#### Question 3:

Divide 24a3b3 by −8ab.

#### Question 4:

Divide −21abc2 by 7abc.

#### Question 5:

Divide 72xyz2 by −9xz.

$\phantom{\rule{0ex}{0ex}}\frac{72xy{z}^{2}}{-9xz}\phantom{\rule{0ex}{0ex}}=\frac{72×x×y×z×z}{-9×x×z}\phantom{\rule{0ex}{0ex}}=-8{x}^{\left(1-1\right)}y{z}^{\left(2-1\right)}\phantom{\rule{0ex}{0ex}}=-8yz$

#### Question 6:

Divide −72a4b5c8 by −9a2b2c3.

$\phantom{\rule{0ex}{0ex}}\frac{-72{a}^{4}{b}^{5}{c}^{8}}{-9{a}^{2}{b}^{2}{c}^{3}}\phantom{\rule{0ex}{0ex}}=\frac{-72×a×a×a×a×b×b×b×b×b×c×c×c×c×c×c×c×c}{-9×a×a×b×b×c×c×c}\phantom{\rule{0ex}{0ex}}=8{a}^{\left(4-2\right)}{b}^{\left(5-2\right)}{c}^{\left(8-3\right)}\phantom{\rule{0ex}{0ex}}=8{a}^{2}{b}^{3}{c}^{5}$

#### Question 7:

Simplify:
$\frac{16{m}^{3}{y}^{2}}{4{m}^{2}y}$

$\phantom{\rule{0ex}{0ex}}\frac{16{m}^{3}{y}^{2}}{4{m}^{2}y}\phantom{\rule{0ex}{0ex}}=\frac{16×m×m×m×y×y}{4×m×m×y}\phantom{\rule{0ex}{0ex}}=4{m}^{\left(3-2\right)}{y}^{\left(2-1\right)}\phantom{\rule{0ex}{0ex}}=4my$

#### Question 8:

Simplify:
$\frac{32{m}^{2}{n}^{3}{p}^{2}}{4mnp}$

$\phantom{\rule{0ex}{0ex}}\frac{32{m}^{2}{n}^{3}{p}^{2}}{4mnp}\phantom{\rule{0ex}{0ex}}=\frac{32×m×m×n×n×n×p×p}{4×m×n×p}\phantom{\rule{0ex}{0ex}}=8{m}^{\left(2-1\right)}{n}^{\left(3-1\right)}{p}^{\left(2-1\right)}\phantom{\rule{0ex}{0ex}}=8m{n}^{2}p$

#### Question 1:

Divide x + 2x2 + 3x4x5 by 2x.

$\phantom{\rule{0ex}{0ex}}\frac{x+2{x}^{2}+3{x}^{4}-{x}^{5}}{2x}\phantom{\rule{0ex}{0ex}}=\frac{x}{2x}+\frac{2{x}^{2}}{2x}+\frac{3{x}^{4}}{2x}-\frac{{x}^{5}}{2x}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}+x+\frac{3}{2}{x}^{3}-\frac{1}{2}{x}^{\mathit{4}}\phantom{\rule{0ex}{0ex}}$

#### Question 2:

Divide .

$\phantom{\rule{0ex}{0ex}}\frac{{y}^{\mathit{4}}\mathit{-}\mathit{3}{y}^{\mathit{3}}\mathit{+}\frac{\mathit{1}}{\mathit{2}}{y}^{\mathit{2}}}{\mathit{3}y}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{{y}^{\mathit{4}}}{\mathit{3}y}\mathit{-}\frac{\mathit{3}{y}^{\mathit{3}}}{\mathit{3}y}\mathit{+}\frac{\frac{\mathit{1}}{\mathit{2}}{y}^{\mathit{2}}}{\mathit{3}y}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{1}}{\mathit{3}}{y}^{\mathit{\left(}\mathit{4}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{-}{y}^{\mathit{\left(}\mathit{3}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{+}\frac{\mathit{1}}{\mathit{6}}{y}^{\mathit{\left(}\mathit{2}\mathit{-}\mathit{1}\mathit{\right)}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{1}}{\mathit{3}}{y}^{\mathit{3}}\mathit{-}{y}^{\mathit{2}}\mathit{+}\frac{\mathit{1}}{\mathit{6}}y$

#### Question 3:

Divide −4a3 + 4a2 + a by 2a.

$\phantom{\rule{0ex}{0ex}}\frac{\mathit{-}\mathit{4}{a}^{\mathit{3}}\mathit{+}\mathit{4}{a}^{\mathit{2}}\mathit{+}a}{\mathit{2}a}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{-}\mathit{4}{a}^{\mathit{3}}}{\mathit{2}a}\mathit{+}\frac{\mathit{4}{a}^{\mathit{2}}}{\mathit{2}a}\mathit{+}\frac{a}{\mathit{2}a}\phantom{\rule{0ex}{0ex}}\mathit{=}\mathit{-}\mathit{2}{a}^{\mathit{\left(}\mathit{3}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{+}\mathit{2}{a}^{\mathit{\left(}\mathit{2}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{+}\frac{\mathit{1}}{\mathit{2}}\phantom{\rule{0ex}{0ex}}\mathit{=}\mathit{-}\mathit{2}{a}^{\mathit{2}}\mathit{+}\mathit{2}a\mathit{+}\frac{\mathit{1}}{\mathit{2}}\phantom{\rule{0ex}{0ex}}$

#### Question 4:

Divide .

$\phantom{\rule{0ex}{0ex}}\frac{\mathit{-}{x}^{\mathit{6}}\mathit{+}\mathit{2}{x}^{\mathit{4}}\mathit{+}\mathit{4}{x}^{\mathit{3}}\mathit{+}\mathit{2}{x}^{\mathit{2}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{-}{x}^{\mathit{6}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\mathit{+}\frac{\mathit{2}{x}^{\mathit{4}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\mathit{+}\frac{\mathit{4}{x}^{\mathit{3}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\mathit{+}\frac{\mathit{2}{x}^{\mathit{2}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{-}\mathit{1}}{\sqrt{\mathit{2}}}{x}^{\mathit{\left(}\mathit{6}\mathit{-}\mathit{2}\mathit{\right)}}\mathit{+}\sqrt{\mathit{2}}{x}^{\mathit{\left(}\mathit{4}\mathit{-}\mathit{2}\mathit{\right)}}\mathit{+}\mathit{2}\sqrt{\mathit{2}}{x}^{\mathit{\left(}\mathit{3}\mathit{-}\mathit{2}\mathit{\right)}}\mathit{+}\sqrt{\mathit{2}}{x}^{\mathit{\left(}\mathit{2}\mathit{-}\mathit{2}\mathit{\right)}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{-}\mathit{1}}{\sqrt{\mathit{2}}}{x}^{\mathit{4}}\mathit{+}\sqrt{\mathit{2}}{x}^{\mathit{2}}\mathit{+}\mathit{2}\sqrt{\mathit{2}}x\mathit{+}\sqrt{\mathit{2}}\phantom{\rule{0ex}{0ex}}$

#### Question 5:

Divide 5z3 − 6z2 + 7z by 2z.

$\phantom{\rule{0ex}{0ex}}\frac{\mathit{5}{z}^{\mathit{3}}\mathit{-}\mathit{6}{z}^{\mathit{2}}\mathit{+}\mathit{7}z}{\mathit{2}z}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{5}{z}^{\mathit{3}}}{\mathit{2}z}\mathit{-}\frac{\mathit{6}{z}^{\mathit{2}}}{\mathit{2}z}\mathit{+}\frac{\mathit{7}z}{\mathit{2}z}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{5}}{\mathit{2}}{z}^{\mathit{\left(}\mathit{3}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{-}\mathit{3}{z}^{\mathit{\left(}\mathit{2}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{+}\frac{\mathit{7}}{\mathit{2}}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{5}}{\mathit{2}}{z}^{\mathit{2}}\mathit{-}\mathit{3}z\mathit{+}\frac{\mathit{7}}{\mathit{2}}$

#### Question 6:

Divide .

$\phantom{\rule{0ex}{0ex}}\frac{\sqrt{\mathit{3}}{a}^{\mathit{4}}\mathit{+}\mathit{2}\sqrt{\mathit{3}}{a}^{\mathit{3}}\mathit{+}\mathit{3}{a}^{\mathit{2}}\mathit{-}\mathit{6}a}{\mathit{3}a}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\sqrt{\mathit{3}}{a}^{\mathit{4}}}{\mathit{3}a}\mathit{+}\frac{\mathit{2}\sqrt{\mathit{3}}{a}^{\mathit{3}}}{\mathit{3}a}\mathit{+}\frac{\mathit{3}{a}^{\mathit{2}}}{\mathit{3}a}\mathit{-}\frac{\mathit{6}a}{\mathit{3}a}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{1}}{\sqrt{\mathit{3}}}{a}^{\mathit{\left(}\mathit{4}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{+}\frac{\mathit{2}}{\sqrt{\mathit{3}}}{a}^{\mathit{\left(}\mathit{3}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{+}{a}^{\mathit{\left(}\mathit{2}\mathit{-}\mathit{1}\mathit{\right)}}\mathit{-}\mathit{2}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{\mathit{1}}{\sqrt{\mathit{3}}}{a}^{\mathit{3}}\mathit{+}\frac{\mathit{2}}{\sqrt{\mathit{3}}}{a}^{\mathit{2}}\mathit{+}a\mathit{-}\mathit{2}$