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Chapter 13

Fundamental Concepts of Algebra — Exercise 13(C)

Class - 7 Concise Mathematics Selina

Exercise 13(C)

Question 1(i)

Multiply:

3x,5x2y3x, 5x^2y and 2y2y

Answer

Solving,

⇒3x×5x2y×2y⇒(3×5×2)×(x×x2×y×y)⇒30x3y2\Rightarrow 3x \times 5x^2y \times 2y\\[1em] \Rightarrow (3 \times 5 \times 2) \times (x \times x^2 \times y \times y)\\[1em] \Rightarrow 30x^3y^2

Hence, 3x×5x2y×2y=30x3y2\bm{3x \times 5x^2y \times 2y = 30x^3y^2}.

Question 1(ii)

Multiply:

5,3a5, 3a and 2ab22ab^2

Answer

Solving,

⇒5×3a×2ab2⇒(5×3×2)×(a×a×b2)⇒30a2b2\Rightarrow 5 \times 3a \times 2ab^2 \\[1em] \Rightarrow (5 \times 3 \times 2) \times (a \times a \times b^2) \\[1em] \Rightarrow 30a^2b^2

Hence, 5×3a×2ab2=30a2b2\bm{5 \times 3a \times 2ab^2 = 30a^2b^2}.

Question 1(iii)

Multiply:

5x+2y5x + 2y and 3xy3xy

Answer

Solving,

⇒(5x+2y)×3xy⇒5x×3xy+2y×3xy⇒15x2y+6xy2\Rightarrow (5x + 2y) \times 3xy \\[1em] \Rightarrow 5x \times 3xy + 2y \times 3xy \\[1em] \Rightarrow 15x^2y + 6xy^2

Hence, (5x+2y)×3xy=15x2y+6xy2\bm{(5x + 2y) \times 3xy = 15x^2y + 6xy^2}.

Question 1(iv)

Multiply:

6a−5b6a - 5b and −2a-2a

Answer

Solving,

⇒(6a−5b)×(−2a)⇒6a×(−2a)−5b×(−2a)⇒−12a2+10ab\Rightarrow (6a - 5b) \times (-2a) \\[1em] \Rightarrow 6a \times (-2a) - 5b \times (-2a) \\[1em] \Rightarrow -12a^2 + 10ab

Hence, (6a−5b)×(−2a)=−12a2+10ab\bm{(6a - 5b) \times (-2a) = -12a^2 + 10ab}.

Question 1(v)

Multiply:

4a+5b4a + 5b and 4a−5b4a - 5b

Answer

Solving,

⇒(4a+5b)(4a−5b)⇒4a(4a−5b)+5b(4a−5b)⇒16a2−20ab+20ab−25b2⇒16a2−25b2\Rightarrow (4a + 5b)(4a - 5b) \\[1em] \Rightarrow 4a(4a - 5b) + 5b(4a - 5b) \\[1em] \Rightarrow 16a^2 - 20ab + 20ab - 25b^2 \\[1em] \Rightarrow 16a^2 - 25b^2

Hence, (4a+5b)(4a−5b)=16a2−25b2\bm{(4a + 5b)(4a - 5b) = 16a^2 - 25b^2}.

Question 1(vi)

Multiply:

9xy+2y29xy + 2y^2 and 2x−3y2x - 3y

Answer

Solving,

⇒(9xy+2y2)(2x−3y)⇒9xy(2x−3y)+2y2(2x−3y)⇒18x2y−27xy2+4xy2−6y3⇒18x2y−23xy2−6y3\Rightarrow (9xy + 2y^2)(2x - 3y) \\[1em] \Rightarrow 9xy(2x - 3y) + 2y^2(2x - 3y) \\[1em] \Rightarrow 18x^2y - 27xy^2 + 4xy^2 - 6y^3 \\[1em] \Rightarrow 18x^2y - 23xy^2 - 6y^3

Hence, (9xy+2y2)(2x−3y)=18x2y−23xy2−6y3\bm{(9xy + 2y^2)(2x - 3y) = 18x^2y - 23xy^2 - 6y^3}.

Question 1(vii)

Multiply:

−3m2n+5mn−4mn2-3m^2n + 5mn - 4mn^2 and 6m2n6m^2n

Answer

Solving,

⇒(−3m2n+5mn−4mn2)×6m2n⇒−3m2n×6m2n+5mn×6m2n−4mn2×6m2n⇒−18m4n2+30m3n2−24m3n3\Rightarrow (-3m^2n + 5mn - 4mn^2) \times 6m^2n \\[1em] \Rightarrow -3m^2n \times 6m^2n + 5mn \times 6m^2n - 4mn^2 \times 6m^2n \\[1em] \Rightarrow -18m^4n^2 + 30m^3n^2 - 24m^3n^3

Hence, (−3m2n+5mn−4mn2)×6m2n=−18m4n2+30m3n2−24m3n3\bm{(-3m^2n + 5mn - 4mn^2) \times 6m^2n = -18m^4n^2 + 30m^3n^2 - 24m^3n^3}.

Question 1(viii)

Multiply:

6xy2−7x2y2+10x36xy^2 - 7x^2y^2 + 10x^3 and −3x2y3-3x^2y^3

Answer

Solving,

⇒(6xy2−7x2y2+10x3)×(−3x2y3)⇒6xy2×(−3x2y3)−7x2y2×(−3x2y3)+10x3×(−3x2y3)⇒−18x3y5+21x4y5−30x5y3\Rightarrow (6xy^2 - 7x^2y^2 + 10x^3) \times (-3x^2y^3) \\[1em] \Rightarrow 6xy^2 \times (-3x^2y^3) - 7x^2y^2 \times (-3x^2y^3) + 10x^3 \times (-3x^2y^3) \\[1em] \Rightarrow -18x^3y^5 + 21x^4y^5 - 30x^5y^3

Hence, (6xy2−7x2y2+10x3)×(−3x2y3)=−18x3y5+21x4y5−30x5y3\bm{(6xy^2 - 7x^2y^2 + 10x^3) \times (-3x^2y^3) = -18x^3y^5 + 21x^4y^5 - 30x^5y^3}.

Question 2(i)

Copy and complete the following multiplication:

3a+2b×−3xy\begin{array}{rrrr} & 3a & + & 2b \\ \times & & - & 3xy \\ \hline \end{array}

Answer

3a+2b×−3xy−9axy−6bxy\begin{array}{rrrr} & 3a & + & 2b \\ \times & & - & 3xy \\ \hline - & 9axy & - & 6bxy \\ \hline \end{array}

Hence, (3a+2b)(−3xy)=−9axy−6bxy\bm{(3a + 2b)(-3xy) = -9axy - 6bxy}.

Question 2(ii)

Copy and complete the following multiplication:

9x−5y×−3xy\begin{array}{rrrr} & 9x & - & 5y \\ \times & & - & 3xy \\ \hline \end{array}

Answer

9x−5y×−3xy−27x2y+15xy2\begin{array}{rrrr} & 9x & - & 5y \\ \times & & - & 3xy \\ \hline - & 27x^2y & + & 15xy^2 \\ \hline \end{array}

Hence, (9x−5y)(−3xy)=−27x2y+15xy2\bm{(9x - 5y)(-3xy) = -27x^2y + 15xy^2}.

Question 2(iii)

Copy and complete the following multiplication:

3xy−2x2−6x×−5x2y\begin{array}{rrrrrr} & 3xy & - & 2x^2 & - & 6x \\ \times & & & & - & 5x^2y \\ \hline \end{array}

Answer

3xy−2x2−6x×−5x2y−15x3y2+10x4y+30x3y\begin{array}{rrrrrr} & 3xy & - & 2x^2 & - & 6x \\ \times & & & & - & 5x^2y \\ \hline - & 15x^3y^2 & + & 10x^4y & + & 30x^3y \\ \hline \end{array}

Hence, (3xy−2x2−6x)(−5x2y)=−15x3y2+10x4y+30x3y\bm{(3xy - 2x^2 - 6x)(-5x^2y) = -15x^3y^2 + 10x^4y + 30x^3y}.

Question 2(iv)

Copy and complete the following multiplication:

a+b×a+b\begin{array}{rrrrrr} & & & a & + & b \\ \times & & & a & + & b \\ \hline \end{array}

Answer

a+b×a+ba2+ab+ab+b2a2+2ab+b2\begin{array}{rrrrrrrr} & & & a & + & b & & \\ \times & & & a & + & b & & \\ \hline & & & a^2 & + & ab & & \\ & & + & & & ab & + & b^2 \\ \hline & & & a^2 & + & 2ab & + & b^2 \\ \hline \end{array}

Hence, (a+b)(a+b)=a2+2ab+b2\bm{(a + b)(a + b) = a^2 + 2ab + b^2}.

Question 2(v)

Copy and complete the following multiplication:

ax−b×2ax+2b2\begin{array}{rrrrrrrr} & & & & & ax & - & b \\ \times & & & & & 2ax & + & 2b^2 \\ \hline \end{array}

Answer

ax−b×2ax+2b22a2x2−2abx+2ab2x−2b32a2x2−2abx+2ab2x−2b3\begin{array}{rrrrrrrr} & & & & & ax & - & b \\ \times & & & & & 2ax & + & 2b^2 \\ \hline & & & & & 2a^2x^2 & - & 2abx \\ & & & & + & 2ab^2x & - & 2b^3 \\ \hline & 2a^2x^2 & - & 2abx & + & 2ab^2x & - & 2b^3 \\ \hline \end{array}

Hence, (ax−b)(2ax+2b2)=2a2x2−2abx+2ab2x−2b3\bm{(ax - b)(2ax + 2b^2) = 2a^2x^2 - 2abx + 2ab^2x - 2b^3}.

Question 2(vi)

Copy and complete the following multiplication:

2a−b+3c×2a−4b\begin{array}{rrrrrrrrrr} & & & & & 2a & - & b & + & 3c \\ \times & & & & & & & 2a & - & 4b \\ \hline \end{array}

Answer

2a−b+3c×2a−4b4a2−2ab+6ac−8ab+4b2−12bc4a2−10ab+6ac+4b2−12bc\begin{array}{rrrrrrrrrr} & & & & & 2a & - & b & + & 3c \\ \times & & & & & & & 2a & - & 4b \\ \hline & & & & & 4a^2 & - & 2ab & + & 6ac \\ & & & & - & 8ab & + & 4b^2 & - & 12bc \\ \hline & 4a^2 & - & 10ab & + & 6ac & + & 4b^2 & - & 12bc \\ \hline \end{array}

Hence, (2a−b+3c)(2a−4b)=4a2−10ab+6ac+4b2−12bc\bm{(2a - b + 3c)(2a - 4b) = 4a^2 - 10ab + 6ac + 4b^2 - 12bc}.

Question 2(vii)

Copy and complete the following multiplication:

3m2+6m−2n×5n−3m\begin{array}{rrrrrrrrrr} & & & & & 3m^2 & + & 6m & - & 2n \\ \times & & & & & & & 5n & - & 3m \\ \hline \end{array}

Answer

3m2+6m−2n×5n−3m15m2n+30mn−10n2−9m3−18m2+6mn−9m3−18m2+15m2n+36mn−10n2\begin{array}{rrrrrrrrrr} & & & & & 3m^2 & + & 6m & - & 2n \\ \times & & & & & & & 5n & - & 3m \\ \hline & & & & & 15m^2n & + & 30mn & - & 10n^2 \\ & & & & - & 9m^3 & - & 18m^2 & + & 6mn \\ \hline - & 9m^3 & - & 18m^2 & + & 15m^2n & + & 36mn & - & 10n^2 \\ \hline \end{array}

Hence, (3m2+6m−2n)(5n−3m)=15m2n+36mn−10n2−9m3−18m2\bm{(3m^2 + 6m - 2n)(5n - 3m) = 15m^2n + 36mn - 10n^2 - 9m^3 - 18m^2}.

Question 2(viii)

Copy and complete the following multiplication:

6−3x+2x2×1+5x−x2\begin{array}{rrrrrrrrrr} & 6 & - & 3x & + & 2x^2 & & & & \\ \times & 1 & + & 5x & - & x^2 & & & & \\ \hline \end{array}

Answer

6−3x+2x2×1+5x−x26−3x+2x2+30x−15x2+10x3−6x2+3x3−2x46+27x−19x2+13x3−2x4\begin{array}{rrrrrrrrrr} & 6 & - & 3x & + & 2x^2 & & & & \\ \times & 1 & + & 5x & - & x^2 & & & & \\ \hline & 6 & - & 3x & + & 2x^2 & & & & \\ + & & & 30x & - & 15x^2 & + & 10x^3 & & \\ - & & & & & 6x^2 & + & 3x^3 & - & 2x^4 \\ \hline & 6 & + & 27x & - & 19x^2 & + & 13x^3 & - & 2x^4 \\ \hline \end{array}

Hence, (6−3x+2x2)(1+5x−x2)=6+27x−19x2+13x3−2x4\bm{(6 - 3x + 2x^2)(1 + 5x - x^2) = 6 + 27x - 19x^2 + 13x^3 - 2x^4}.

Question 2(ix)

Copy and complete the following multiplication:

4x3−10x2+6x−8×−x2+2x+3\begin{array}{rrrrrrrrrrrr} & & & & & 4x^3 & - & 10x^2 & + & 6x & - & 8 \\ \times & & & & & & & -x^2 & + & 2x & + & 3 \\ \hline \end{array}

Answer

4x3−10x2+6x−8×−x2+2x+312x3−30x2+18x−24+8x4−20x3+12x2−16x−4x5+10x4−6x3+8x2−4x5+18x4−14x3−10x2+2x−24\begin{array}{rrrrrrrrrrrr} & & & & & 4x^3 & - & 10x^2 & + & 6x & - & 8 \\ \times & & & & & & & -x^2 & + & 2x & + & 3 \\ \hline & & & & & 12x^3 & - & 30x^2 & + & 18x & - & 24 \\ + & & & 8x^4 & - & 20x^3 & + & 12x^2 & - & 16x & & \\ - & 4x^5 & + & 10x^4 & - & 6x^3 & + & 8x^2 & & & & \\ \hline - & 4x^5 & + & 18x^4 & - & 14x^3 & - & 10x^2 & + & 2x & - & 24 \\ \hline \end{array}

Hence,

(4x3−10x2+6x−8)(3+2x−x2)=−4x5+18x4−14x3−10x2+2x−24.\bm{(4x^3 - 10x^2 + 6x - 8)(3 + 2x - x^2) = -4x^5 + 18x^4 - 14x^3 - 10x^2 + 2x - 24}.

Question 3(i)

Evaluate:

(c+5)(c−3)(c + 5)(c - 3)

Answer

Solving,

⇒(c+5)(c−3)⇒c(c−3)+5(c−3)⇒c2−3c+5c−15⇒c2+2c−15\Rightarrow (c + 5)(c - 3) \\[1em] \Rightarrow c(c - 3) + 5(c - 3) \\[1em] \Rightarrow c^2 - 3c + 5c - 15 \\[1em] \Rightarrow c^2 + 2c - 15

Hence, (c+5)(c−3)=c2+2c−15\bm{(c + 5)(c - 3) = c^2 + 2c - 15}.

Question 3(ii)

Evaluate:

(3c−5d)(4c−6d)(3c - 5d)(4c - 6d)

Answer

Solving,

⇒(3c−5d)(4c−6d)⇒3c(4c−6d)−5d(4c−6d)⇒12c2−18cd−20cd+30d2⇒12c2−38cd+30d2\Rightarrow (3c - 5d)(4c - 6d) \\[1em] \Rightarrow 3c(4c - 6d) - 5d(4c - 6d) \\[1em] \Rightarrow 12c^2 - 18cd - 20cd + 30d^2 \\[1em] \Rightarrow 12c^2 - 38cd + 30d^2

Hence, (3c−5d)(4c−6d)=12c2−38cd+30d2\bm{(3c - 5d)(4c - 6d) = 12c^2 - 38cd + 30d^2}.

Question 3(iii)

Evaluate:

(12a+12b)(12a−12b)\left(\dfrac{1}{2}a + \dfrac{1}{2}b\right)\left(\dfrac{1}{2}a - \dfrac{1}{2}b\right)

Answer

Solving,

⇒(12a+12b)(12a−12b)⇒12a(12a−12b)+12b(12a−12b)⇒14a2−14ab+14ab−14b2⇒14a2−14b2\Rightarrow \left(\dfrac{1}{2}a + \dfrac{1}{2}b\right)\left(\dfrac{1}{2}a - \dfrac{1}{2}b\right) \\[1em] \Rightarrow \dfrac{1}{2}a\left(\dfrac{1}{2}a - \dfrac{1}{2}b\right) + \dfrac{1}{2}b\left(\dfrac{1}{2}a - \dfrac{1}{2}b\right) \\[1em] \Rightarrow \dfrac{1}{4}a^2 - \dfrac{1}{4}ab + \dfrac{1}{4}ab - \dfrac{1}{4}b^2 \\[1em] \Rightarrow \dfrac{1}{4}a^2 - \dfrac{1}{4}b^2

Hence, (12a+12b)(12a−12b)=14a2−14b2\bm{\left(\dfrac{1}{2}a + \dfrac{1}{2}b\right)\left(\dfrac{1}{2}a - \dfrac{1}{2}b\right) = \dfrac{1}{4}a^2 - \dfrac{1}{4}b^2}.

Question 3(iv)

Evaluate:

(a2+2ab+b2)(a+b)(a^2 + 2ab + b^2)(a + b)

Answer

Solving,

⇒(a2+2ab+b2)(a+b)⇒a(a2+2ab+b2)+b(a2+2ab+b2)⇒a3+2a2b+ab2+a2b+2ab2+b3⇒a3+3a2b+3ab2+b3\Rightarrow (a^2 + 2ab + b^2)(a + b) \\[1em] \Rightarrow a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2) \\[1em] \Rightarrow a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3 \\[1em] \Rightarrow a^3 + 3a^2b + 3ab^2 + b^3

Hence, (a2+2ab+b2)(a+b)=a3+3a2b+3ab2+b3\bm{(a^2 + 2ab + b^2)(a + b) = a^3 + 3a^2b + 3ab^2 + b^3}.

Question 3(v)

Evaluate:

(3x−1)(4x3−2x2+6x−3)(3x - 1)(4x^3 - 2x^2 + 6x - 3)

Answer

Solving,

⇒(3x−1)(4x3−2x2+6x−3)⇒3x(4x3−2x2+6x−3)−1(4x3−2x2+6x−3)⇒12x4−6x3+18x2−9x−4x3+2x2−6x+3⇒12x4−10x3+20x2−15x+3\Rightarrow (3x - 1)(4x^3 - 2x^2 + 6x - 3) \\[1em] \Rightarrow 3x(4x^3 - 2x^2 + 6x - 3) - 1(4x^3 - 2x^2 + 6x - 3) \\[1em] \Rightarrow 12x^4 - 6x^3 + 18x^2 - 9x - 4x^3 + 2x^2 - 6x + 3 \\[1em] \Rightarrow 12x^4 - 10x^3 + 20x^2 - 15x + 3

Hence, (3x−1)(4x3−2x2+6x−3)=12x4−10x3+20x2−15x+3\bm{(3x - 1)(4x^3 - 2x^2 + 6x - 3) = 12x^4 - 10x^3 + 20x^2 - 15x + 3}.

Question 3(vi)

Evaluate:

(4m−2)(m2+5m−6)(4m - 2)(m^2 + 5m - 6)

Answer

Solving,

⇒(4m−2)(m2+5m−6)⇒4m(m2+5m−6)−2(m2+5m−6)⇒4m3+20m2−24m−2m2−10m+12⇒4m3+18m2−34m+12\Rightarrow (4m - 2)(m^2 + 5m - 6) \\[1em] \Rightarrow 4m(m^2 + 5m - 6) - 2(m^2 + 5m - 6) \\[1em] \Rightarrow 4m^3 + 20m^2 - 24m - 2m^2 - 10m + 12 \\[1em] \Rightarrow 4m^3 + 18m^2 - 34m + 12

Hence, (4m−2)(m2+5m−6)=4m3+18m2−34m+12\bm{(4m - 2)(m^2 + 5m - 6) = 4m^3 + 18m^2 - 34m + 12}.

Question 3(vii)

Evaluate:

(8−12x+7x2−6x3)(5−2x)(8 - 12x + 7x^2 - 6x^3)(5 - 2x)

Answer

Solving,

⇒(8−12x+7x2−6x3)(5−2x)⇒5(8−12x+7x2−6x3)−2x(8−12x+7x2−6x3)⇒40−60x+35x2−30x3−16x+24x2−14x3+12x4⇒40−76x+59x2−44x3+12x4\Rightarrow (8 - 12x + 7x^2 - 6x^3)(5 - 2x) \\[1em] \Rightarrow 5(8 - 12x + 7x^2 - 6x^3) - 2x(8 - 12x + 7x^2 - 6x^3) \\[1em] \Rightarrow 40 - 60x + 35x^2 - 30x^3 - 16x + 24x^2 - 14x^3 + 12x^4 \\[1em] \Rightarrow 40 - 76x + 59x^2 - 44x^3 + 12x^4

Hence, (8−12x+7x2−6x3)(5−2x)=40−76x+59x2−44x3+12x4\bm{(8 - 12x + 7x^2 - 6x^3)(5 - 2x) = 40 - 76x + 59x^2 - 44x^3 + 12x^4}.

Question 3(viii)

Evaluate:

(4x2−4x+1)(2x3−3x2+2)(4x^2 - 4x + 1)(2x^3 - 3x^2 + 2)

Answer

Solving,

⇒(4x2−4x+1)(2x3−3x2+2)⇒4x2(2x3−3x2+2)−4x(2x3−3x2+2)+1(2x3−3x2+2)⇒8x5−12x4+8x2−8x4+12x3−8x+2x3−3x2+2⇒8x5+(−12−8)x4+(12+2)x3+(8−3)x2−8x+2⇒8x5−20x4+14x3+5x2−8x+2\Rightarrow (4x^2 - 4x + 1)(2x^3 - 3x^2 + 2) \\[1em] \Rightarrow 4x^2(2x^3 - 3x^2 + 2) - 4x(2x^3 - 3x^2 + 2) + 1(2x^3 - 3x^2 + 2) \\[1em] \Rightarrow 8x^5 - 12x^4 + 8x^2 - 8x^4 + 12x^3 - 8x + 2x^3 - 3x^2 + 2 \\[1em] \Rightarrow 8x^5 + (-12 - 8)x^4 + (12 + 2)x^3 + (8 - 3)x^2 - 8x + 2 \\[1em] \Rightarrow 8x^5 - 20x^4 + 14x^3 + 5x^2 - 8x + 2

Hence, (4x2−4x+1)(2x3−3x2+2)=8x5−20x4+14x3+5x2−8x+2\bm{(4x^2 - 4x + 1)(2x^3 - 3x^2 + 2) = 8x^5 - 20x^4 + 14x^3 + 5x^2 - 8x + 2}.

Question 3(ix)

Evaluate:

(6p2−8pq+2q2)(−5p)(6p^2 - 8pq + 2q^2)(- 5p)

Answer

Solving,

⇒(6p2−8pq+2q2)×(−5p)⇒6p2×(−5p)−8pq×(−5p)+2q2×(−5p)⇒−30p3+40p2q−10pq2\Rightarrow (6p^2 - 8pq + 2q^2) \times (-5p) \\[1em] \Rightarrow 6p^2 \times (-5p) - 8pq \times (-5p) + 2q^2 \times (-5p) \\[1em] \Rightarrow -30p^3 + 40p^2q - 10pq^2

Hence, (6p2−8pq+2q2)×(−5p)=−30p3+40p2q−10pq2\bm{(6p^2 - 8pq + 2q^2) \times (-5p) = -30p^3 + 40p^2q - 10pq^2}.

Question 3(x)

Evaluate:

−4y(15x+12y−8z)(x−2y)-4y (15x + 12y - 8z)(x - 2y)

Answer

Solving,

⇒−4y(15x+12y−8z)(x−2y)⇒(−60xy−48y2+32yz)(x−2y)⇒−60xy(x−2y)−48y2(x−2y)+32yz(x−2y)⇒−60x2y+120xy2−48xy2+96y3+32xyz−64y2z⇒−60x2y+72xy2+32xyz+96y3−64y2z\Rightarrow -4y(15x + 12y - 8z)(x - 2y) \\[1em] \Rightarrow (-60xy - 48y^2 + 32yz)(x - 2y) \\[1em] \Rightarrow -60xy(x - 2y) - 48y^2(x - 2y) + 32yz(x - 2y) \\[1em] \Rightarrow -60x^2y + 120xy^2 - 48xy^2 + 96y^3 + 32xyz - 64y^2z \\[1em] \Rightarrow -60x^2y + 72xy^2 + 32xyz + 96y^3 - 64y^2z

Hence, −4y(15x+12y−8z)(x−2y)=−60x2y+72xy2+32xyz+96y3−64y2z\bm{-4y(15x + 12y - 8z)(x - 2y) = -60x^2y + 72xy^2 + 32xyz + 96y^3 - 64y^2z}.

Question 3(xi)

Evaluate:

(a2+b2+c2−ab−bc−ca)(a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)(a + b + c)

Answer

Solving,

⇒(a2+b2+c2−ab−bc−ca)(a+b+c)⇒a(a2+b2+c2−ab−bc−ca)+b(a2+b2+c2−ab−bc−ca)+c(a2+b2+c2−ab−bc−ca)⇒a3+ab2+ac2−a2b−abc−a2c+a2b+b3+bc2−ab2−b2c−abc+a2c+b2c+c3−abc−bc2−ac2⇒a3+b3+c3−3abc\Rightarrow (a^2 + b^2 + c^2 - ab - bc - ca)(a + b + c) \\[1em] \Rightarrow a(a^2 + b^2 + c^2 - ab - bc - ca) + b(a^2 + b^2 + c^2 - ab - bc - ca) + c(a^2 + b^2 + c^2 - ab - bc - ca) \\[1em] \Rightarrow a^3 + ab^2 + ac^2 - a^2b - abc - a^2c + a^2b + b^3 + bc^2 - ab^2 - b^2c - abc + a^2c + b^2c + c^3 - abc - bc^2 - ac^2 \\[1em] \Rightarrow a^3 + b^3 + c^3 - 3abc

Hence, (a2+b2+c2−ab−bc−ca)(a+b+c)=a3+b3+c3−3abc\bm{(a^2 + b^2 + c^2 - ab - bc - ca)(a + b + c) = a^3 + b^3 + c^3 - 3abc}.

Question 4

Evaluate:

(i) (a+b)(a−b)(a + b)(a - b).

(ii) (a2+b2)(a+b)(a−b)(a^2 + b^2)(a + b)(a - b), using the result of (i).

(iii) (a4+b4)(a2+b2)(a+b)(a−b)(a^4 + b^4)(a^2 + b^2)(a + b)(a - b), using the result of (ii).

Answer

(i) (a+b)(a−b)(a + b)(a - b)

⇒a(a−b)+b(a−b)⇒a2−ab+ab−b2⇒a2−b2\Rightarrow a(a - b) + b(a - b) \\[1em] \Rightarrow a^2 - ab + ab - b^2 \\[1em] \Rightarrow a^2 - b^2

Hence, (a+b)(a−b)=a2−b2\bm{(a + b)(a - b) = a^2 - b^2}.

(ii) (a2+b2)(a+b)(a−b)(a^2 + b^2)(a + b)(a - b)

⇒(a2+b2)(a2−b2)[Using the result of (i)]⇒a2(a2−b2)+b2(a2−b2)⇒a4−a2b2+a2b2−b4⇒a4−b4\Rightarrow (a^2 + b^2)(a^2 - b^2) \quad \text{[Using the result of (i)]} \\[1em] \Rightarrow a^2(a^2 - b^2) + b^2(a^2 - b^2) \\[1em] \Rightarrow a^4 - a^2b^2 + a^2b^2 - b^4 \\[1em] \Rightarrow a^4 - b^4

Hence, (a2+b2)(a+b)(a−b)=a4−b4\bm{(a^2 + b^2)(a + b)(a - b) = a^4 - b^4}.

(iii) (a4+b4)(a2+b2)(a+b)(a−b)(a^4 + b^4)(a^2 + b^2)(a + b)(a - b)

⇒(a4+b4)(a4−b4)[Using the result of (ii)]⇒a4(a4−b4)+b4(a4−b4)⇒a8−a4b4+a4b4−b8⇒a8−b8\Rightarrow (a^4 + b^4)(a^4 - b^4) \quad \text{[Using the result of (ii)]} \\[1em] \Rightarrow a^4(a^4 - b^4) + b^4(a^4 - b^4) \\[1em] \Rightarrow a^8 - a^4b^4 + a^4b^4 - b^8 \\[1em] \Rightarrow a^8 - b^8

Hence, (a4+b4)(a2+b2)(a+b)(a−b)=a8−b8\bm{(a^4 + b^4)(a^2 + b^2)(a + b)(a - b) = a^8 - b^8}.

Question 5(i)

Evaluate:

(3x−2y)(4x+3y)(3x - 2y)(4x + 3y).

Answer

Solving,

⇒(3x−2y)(4x+3y)⇒3x(4x+3y)−2y(4x+3y)⇒12x2+9xy−8xy−6y2⇒12x2+xy−6y2\Rightarrow (3x - 2y)(4x + 3y) \\[1em] \Rightarrow 3x(4x + 3y) - 2y(4x + 3y) \\[1em] \Rightarrow 12x^2 + 9xy - 8xy - 6y^2 \\[1em] \Rightarrow 12x^2 + xy - 6y^2

Hence, (3x−2y)(4x+3y)=12x2+xy−6y2\bm{(3x - 2y)(4x + 3y) = 12x^2 + xy - 6y^2}.

Question 5(ii)

Evaluate:

(3x−2y)(4x+3y)(8x−5y)(3x - 2y)(4x + 3y)(8x - 5y).

Answer

Solving,

⇒(3x−2y)(4x+3y)(8x−5y)⇒(12x2+xy−6y2)(8x−5y)[Using the result of (i)]⇒8x(12x2+xy−6y2)−5y(12x2+xy−6y2)⇒96x3+8x2y−48xy2−60x2y−5xy2+30y3⇒96x3−52x2y−53xy2+30y3\Rightarrow (3x - 2y)(4x + 3y)(8x - 5y) \\[1em] \Rightarrow (12x^2 + xy - 6y^2)(8x - 5y) \quad \text{[Using the result of (i)]} \\[1em] \Rightarrow 8x(12x^2 + xy - 6y^2) - 5y(12x^2 + xy - 6y^2) \\[1em] \Rightarrow 96x^3 + 8x^2y - 48xy^2 - 60x^2y - 5xy^2 + 30y^3 \\[1em] \Rightarrow 96x^3 - 52x^2y - 53xy^2 + 30y^3

Hence, (3x−2y)(4x+3y)(8x−5y)=96x3−52x2y−53xy2+30y3\bm{(3x - 2y)(4x + 3y)(8x - 5y) = 96x^3 - 52x^2y - 53xy^2 + 30y^3}.

Question 5(iii)

Evaluate:

(a+5)(3a−2)(5a+1)(a + 5)(3a - 2)(5a + 1)

Answer

Solving,

⇒(a+5)(3a−2)(5a+1)⇒[a(3a−2)+5(3a−2)](5a+1)⇒(3a2−2a+15a−10)(5a+1)⇒(3a2+13a−10)(5a+1)⇒5a(3a2+13a−10)+1(3a2+13a−10)⇒15a3+65a2−50a+3a2+13a−10⇒15a3+68a2−37a−10\Rightarrow (a + 5)(3a - 2)(5a + 1) \\[1em] \Rightarrow [a(3a - 2) + 5(3a - 2)](5a + 1) \\[1em] \Rightarrow (3a^2 - 2a + 15a - 10)(5a + 1) \\[1em] \Rightarrow (3a^2 + 13a - 10)(5a + 1) \\[1em] \Rightarrow 5a(3a^2 + 13a - 10) + 1(3a^2 + 13a - 10) \\[1em] \Rightarrow 15a^3 + 65a^2 - 50a + 3a^2 + 13a - 10 \\[1em] \Rightarrow 15a^3 + 68a^2 - 37a - 10

Hence, (a+5)(3a−2)(5a+1)=15a3+68a2−37a−10\bm{(a + 5)(3a - 2)(5a + 1) = 15a^3 + 68a^2 - 37a - 10}.

Question 5(iv)

Evaluate:

(a+1)(a2−a+1)(a + 1)(a^2 - a + 1) and (a−1)(a2+a+1)(a - 1)(a^2 + a + 1);

and then: (a+1)(a2−a+1)+(a−1)(a2+a+1)(a + 1)(a^2 - a + 1) + (a - 1)(a^2 + a + 1).

Answer

Solving (a+1)(a2−a+1)(a + 1)(a^2 - a + 1), we get :

⇒(a+1)(a2−a+1)⇒a(a2−a+1)+1(a2−a+1)⇒a3−a2+a+a2−a+1⇒a3+1\Rightarrow (a + 1)(a^2 - a + 1) \\[1em] \Rightarrow a(a^2 - a + 1) + 1(a^2 - a + 1) \\[1em] \Rightarrow a^3 - a^2 + a + a^2 - a + 1 \\[1em] \Rightarrow a^3 + 1

Solving (a−1)(a2+a+1)(a - 1)(a^2 + a + 1), we get :

⇒(a−1)(a2+a+1)⇒a(a2+a+1)−1(a2+a+1)⇒a3+a2+a−a2−a−1⇒a3−1\Rightarrow (a - 1)(a^2 + a + 1) \\[1em] \Rightarrow a(a^2 + a + 1) - 1(a^2 + a + 1) \\[1em] \Rightarrow a^3 + a^2 + a - a^2 - a - 1 \\[1em] \Rightarrow a^3 - 1

Solving, (a+1)(a2−a+1)+(a−1)(a2+a+1)(a + 1)(a^2 - a + 1) + (a - 1)(a^2 + a + 1)

⇒(a+1)(a2−a+1)+(a−1)(a2+a+1)⇒(a3+1)+(a3−1)⇒2a3\Rightarrow (a + 1)(a^2 - a + 1) + (a - 1)(a^2 + a + 1) \\[1em] \Rightarrow (a^3 + 1) + (a^3 - 1) \\[1em] \Rightarrow 2a^3

Hence, (a+1)(a2−a+1)+(a−1)(a2+a+1)=2a3\bm{(a + 1)(a^2 - a + 1) + (a - 1)(a^2 + a + 1) = 2a^3}.

Question 5(v)

Evaluate:

(5m−2n)(5m+2n)(25m2+4n2)(5m - 2n)(5m + 2n)(25m^2 + 4n^2).

Answer

Solving,

⇒(5m−2n)(5m+2n)(25m2+4n2)⇒[5m(5m+2n)−2n(5m+2n)](25m2+4n2)⇒(25m2+10mn−10mn−4n2)(25m2+4n2)⇒(25m2−4n2)(25m2+4n2)⇒25m2(25m2+4n2)−4n2(25m2+4n2)⇒625m4+100m2n2−100m2n2−16n4⇒625m4−16n4\Rightarrow (5m - 2n)(5m + 2n)(25m^2 + 4n^2) \\[1em] \Rightarrow [5m(5m + 2n) - 2n(5m + 2n)](25m^2 + 4n^2) \\[1em] \Rightarrow (25m^2 + 10mn - 10mn - 4n^2)(25m^2 + 4n^2) \\[1em] \Rightarrow (25m^2 - 4n^2)(25m^2 + 4n^2) \\[1em] \Rightarrow 25m^2(25m^2 + 4n^2) - 4n^2(25m^2 + 4n^2) \\[1em] \Rightarrow 625m^4 + 100m^2n^2 - 100m^2n^2 - 16n^4 \\[1em] \Rightarrow 625m^4 - 16n^4

Hence, (5m−2n)(5m+2n)(25m2+4n2)=625m4−16n4\bm{(5m - 2n)(5m + 2n)(25m^2 + 4n^2) = 625m^4 - 16n^4}.

Question 6(i)

Multiply:

mn4,m3nmn^4, m^3n and 5m2n35m^2n^3

Answer

Solving,

⇒mn4×m3n×5m2n3⇒5×(m×m3×m2)×(n4×n×n3)⇒5m6n8\Rightarrow mn^4 \times m^3n \times 5m^2n^3 \\[1em] \Rightarrow 5 \times (m \times m^3 \times m^2) \times (n^4 \times n \times n^3) \\[1em] \Rightarrow 5m^6n^8

Hence, mn4×m3n×5m2n3=5m6n8\bm{mn^4 \times m^3n \times 5m^2n^3 = 5m^6n^8}.

Question 6(ii)

Multiply:

2mnpq,4mnpq2mnpq, 4mnpq and 5mnpq5mnpq

Answer

Solving,

⇒2mnpq×4mnpq×5mnpq⇒(2×4×5)×m3n3p3q3⇒40m3n3p3q3\Rightarrow 2mnpq \times 4mnpq \times 5mnpq \\[1em] \Rightarrow (2 \times 4 \times 5) \times m^3n^3p^3q^3 \\[1em] \Rightarrow 40m^3n^3p^3q^3

Hence, 2mnpq×4mnpq×5mnpq=40m3n3p3q3\bm{2mnpq \times 4mnpq \times 5mnpq = 40m^3n^3p^3q^3}.

Question 6(iii)

Multiply:

pq−pmpq - pm and p2mp^2m

Answer

Solving,

⇒(pq−pm)×p2m⇒pq×p2m−pm×p2m⇒p3qm−p3m2\Rightarrow (pq - pm) \times p^2m \\[1em] \Rightarrow pq \times p^2m - pm \times p^2m \\[1em] \Rightarrow p^3qm - p^3m^2

Hence, (pq−pm)×p2m=p3qm−p3m2\bm{(pq - pm) \times p^2m = p^3qm - p^3m^2}.

Question 6(iv)

Multiply:

x3−3y3x^3 - 3y^3 and 4x2y24x^2y^2

Answer

Solving,

⇒(x3−3y3)×4x2y2⇒x3×4x2y2−3y3×4x2y2⇒4x5y2−12x2y5\Rightarrow (x^3 - 3y^3) \times 4x^2y^2 \\[1em] \Rightarrow x^3 \times 4x^2y^2 - 3y^3 \times 4x^2y^2 \\[1em] \Rightarrow 4x^5y^2 - 12x^2y^5

Hence, (x3−3y3)×4x2y2=4x5y2−12x2y5\bm{(x^3 - 3y^3) \times 4x^2y^2 = 4x^5y^2 - 12x^2y^5}.

Question 6(v)

Multiply:

a3−4aba^3 - 4ab and 2a2b2a^2b

Answer

Solving,

⇒(a3−4ab)×2a2b⇒a3×2a2b−4ab×2a2b⇒2a5b−8a3b2\Rightarrow (a^3 - 4ab) \times 2a^2b \\[1em] \Rightarrow a^3 \times 2a^2b - 4ab \times 2a^2b \\[1em] \Rightarrow 2a^5b - 8a^3b^2

Hence, (a3−4ab)×2a2b=2a5b−8a3b2\bm{(a^3 - 4ab) \times 2a^2b = 2a^5b - 8a^3b^2}.

Question 6(vi)

Multiply:

x2+5yx−3y2x^2 + 5yx - 3y^2 and 2x2y2x^2y.

Answer

Solving,

⇒(x2+5yx−3y2)×2x2y⇒x2×2x2y+5xy×2x2y−3y2×2x2y⇒2x4y+10x3y2−6x2y3\Rightarrow (x^2 + 5yx - 3y^2) \times 2x^2y \\[1em] \Rightarrow x^2 \times 2x^2y + 5xy \times 2x^2y - 3y^2 \times 2x^2y \\[1em] \Rightarrow 2x^4y + 10x^3y^2 - 6x^2y^3

Hence, (x2+5yx−3y2)×2x2y=2x4y+10x3y2−6x2y3\bm{(x^2 + 5yx - 3y^2) \times 2x^2y = 2x^4y + 10x^3y^2 - 6x^2y^3}.

Question 7(i)

Multiply:

(2x+3y)(2x+3y)(2x + 3y) (2x + 3y)

Answer

Solving,

⇒(2x+3y)(2x+3y)⇒2x(2x+3y)+3y(2x+3y)⇒4x2+6xy+6xy+9y2⇒4x2+12xy+9y2\Rightarrow (2x + 3y)(2x + 3y) \\[1em] \Rightarrow 2x(2x + 3y) + 3y(2x + 3y) \\[1em] \Rightarrow 4x^2 + 6xy + 6xy + 9y^2 \\[1em] \Rightarrow 4x^2 + 12xy + 9y^2

Hence, (2x+3y)(2x+3y)=4x2+12xy+9y2\bm{(2x + 3y)(2x + 3y) = 4x^2 + 12xy + 9y^2}.

Question 7(ii)

Multiply:

(2x−3y)(2x+3y)(2x - 3y) (2x + 3y)

Answer

Solving,

⇒(2x−3y)(2x+3y)⇒2x(2x+3y)−3y(2x+3y)⇒4x2+6xy−6xy−9y2⇒4x2−9y2\Rightarrow (2x - 3y)(2x + 3y) \\[1em] \Rightarrow 2x(2x + 3y) - 3y(2x + 3y) \\[1em] \Rightarrow 4x^2 + 6xy - 6xy - 9y^2 \\[1em] \Rightarrow 4x^2 - 9y^2

Hence, (2x−3y)(2x+3y)=4x2−9y2\bm{(2x - 3y)(2x + 3y) = 4x^2 - 9y^2}.

Question 7(iii)

Multiply:

(2x+3y)(2x−3y)(2x + 3y) (2x - 3y)

Answer

Solving,

⇒(2x+3y)(2x−3y)⇒2x(2x−3y)+3y(2x−3y)⇒4x2−6xy+6xy−9y2⇒4x2−9y2\Rightarrow (2x + 3y)(2x - 3y) \\[1em] \Rightarrow 2x(2x - 3y) + 3y(2x - 3y) \\[1em] \Rightarrow 4x^2 - 6xy + 6xy - 9y^2 \\[1em] \Rightarrow 4x^2 - 9y^2

Hence, (2x+3y)(2x−3y)=4x2−9y2\bm{(2x + 3y)(2x - 3y) = 4x^2 - 9y^2}.

Question 7(iv)

Multiply:

(2x−3y)(2x−3y)(2x - 3y) (2x - 3y)

Answer

Solving,

⇒(2x−3y)(2x−3y)⇒2x(2x−3y)−3y(2x−3y)⇒4x2−6xy−6xy+9y2⇒4x2−12xy+9y2\Rightarrow (2x - 3y)(2x - 3y) \\[1em] \Rightarrow 2x(2x - 3y) - 3y(2x - 3y) \\[1em] \Rightarrow 4x^2 - 6xy - 6xy + 9y^2 \\[1em] \Rightarrow 4x^2 - 12xy + 9y^2

Hence, (2x−3y)(2x−3y)=4x2−12xy+9y2\bm{(2x - 3y)(2x - 3y) = 4x^2 - 12xy + 9y^2}.

Question 7(v)

Multiply:

(−2x+3y)(2x−3y)(-2x + 3y) (2x - 3y)

Answer

Solving,

⇒(−2x+3y)(2x−3y)⇒−2x(2x−3y)+3y(2x−3y)⇒−4x2+6xy+6xy−9y2⇒−4x2+12xy−9y2\Rightarrow (-2x + 3y)(2x - 3y) \\[1em] \Rightarrow -2x(2x - 3y) + 3y(2x - 3y) \\[1em] \Rightarrow -4x^2 + 6xy + 6xy - 9y^2 \\[1em] \Rightarrow -4x^2 + 12xy - 9y^2

Hence, (−2x+3y)(2x−3y)=−4x2+12xy−9y2\bm{(-2x + 3y)(2x - 3y) = -4x^2 + 12xy - 9y^2}.

Question 7(vi)

Multiply:

(xy+2b)(xy−2b)(xy + 2b) (xy - 2b)

Answer

Solving,

⇒(xy+2b)(xy−2b)⇒xy(xy−2b)+2b(xy−2b)⇒x2y2−2bxy+2bxy−4b2⇒x2y2−4b2\Rightarrow (xy + 2b)(xy - 2b) \\[1em] \Rightarrow xy(xy - 2b) + 2b(xy - 2b) \\[1em] \Rightarrow x^2y^2 - 2bxy + 2bxy - 4b^2 \\[1em] \Rightarrow x^2y^2 - 4b^2

Hence, (xy+2b)(xy−2b)=x2y2−4b2\bm{(xy + 2b)(xy - 2b) = x^2y^2 - 4b^2}.

Question 7(vii)

Multiply:

(x−a)(x+3b)(x - a) (x + 3b)

Answer

Solving,

⇒(x−a)(x+3b)⇒x(x+3b)−a(x+3b)⇒x2+3bx−ax−3ab\Rightarrow (x - a)(x + 3b) \\[1em] \Rightarrow x(x + 3b) - a(x + 3b) \\[1em] \Rightarrow x^2 + 3bx - ax - 3ab

Hence, (x−a)(x+3b)=x2+3bx−ax−3ab\bm{(x - a)(x + 3b) = x^2 + 3bx - ax - 3ab}.

Question 7(viii)

Multiply:

(2x+5y+6)(3x+y−8)(2x + 5y + 6) (3x + y - 8)

Answer

Solving,

⇒(2x+5y+6)(3x+y−8)⇒2x(3x+y−8)+5y(3x+y−8)+6(3x+y−8)⇒6x2+2xy−16x+15xy+5y2−40y+18x+6y−48⇒6x2+(2+15)xy+5y2+(−16+18)x+(−40+6)y−48⇒6x2+17xy+5y2+2x−34y−48\Rightarrow (2x + 5y + 6)(3x + y - 8) \\[1em] \Rightarrow 2x(3x + y - 8) + 5y(3x + y - 8) + 6(3x + y - 8) \\[1em] \Rightarrow 6x^2 + 2xy - 16x + 15xy + 5y^2 - 40y + 18x + 6y - 48 \\[1em] \Rightarrow 6x^2 + (2 + 15)xy + 5y^2 + (-16 + 18)x + (-40 + 6)y - 48 \\[1em] \Rightarrow 6x^2 + 17xy + 5y^2 + 2x - 34y - 48

Hence, (2x+5y+6)(3x+y−8)=6x2+17xy+5y2+2x−34y−48\bm{(2x + 5y + 6)(3x + y - 8) = 6x^2 + 17xy + 5y^2 + 2x - 34y - 48}.

Question 7(ix)

Multiply:

(3x−5y+2)(5x−4y−3)(3x - 5y + 2) (5x - 4y - 3)

Answer

Solving,

⇒(3x−5y+2)(5x−4y−3)⇒3x(5x−4y−3)−5y(5x−4y−3)+2(5x−4y−3)⇒15x2−12xy−9x−25xy+20y2+15y+10x−8y−6⇒15x2+(−12−25)xy+20y2+(−9+10)x+(15−8)y−6⇒15x2−37xy+20y2+x+7y−6\Rightarrow (3x - 5y + 2)(5x - 4y - 3) \\[1em] \Rightarrow 3x(5x - 4y - 3) - 5y(5x - 4y - 3) + 2(5x - 4y - 3) \\[1em] \Rightarrow 15x^2 - 12xy - 9x - 25xy + 20y^2 + 15y + 10x - 8y - 6 \\[1em] \Rightarrow 15x^2 + (-12 - 25)xy + 20y^2 + (-9 + 10)x + (15 - 8)y - 6 \\[1em] \Rightarrow 15x^2 - 37xy + 20y^2 + x + 7y - 6

Hence, (3x−5y+2)(5x−4y−3)=15x2−37xy+20y2+x+7y−6\bm{(3x - 5y + 2)(5x - 4y - 3) = 15x^2 - 37xy + 20y^2 + x + 7y - 6}.

Question 7(x)

Multiply:

(6x−2y)(3x−y)(6x - 2y) (3x - y)

Answer

Solving,

⇒(6x−2y)(3x−y)⇒6x(3x−y)−2y(3x−y)⇒18x2−6xy−6xy+2y2⇒18x2−12xy+2y2\Rightarrow (6x - 2y)(3x - y) \\[1em] \Rightarrow 6x(3x - y) - 2y(3x - y) \\[1em] \Rightarrow 18x^2 - 6xy - 6xy + 2y^2 \\[1em] \Rightarrow 18x^2 - 12xy + 2y^2

Hence, (6x−2y)(3x−y)=18x2−12xy+2y2\bm{(6x - 2y)(3x - y) = 18x^2 - 12xy + 2y^2}.

Question 7(xi)

Multiply:

(1+6x2−4x3)(−1+3x−3x2)(1 + 6x^2 - 4x^3) (-1 + 3x - 3x^2)

Answer

Solving,

⇒(1+6x2−4x3)(−1+3x−3x2)⇒1(−1+3x−3x2)+6x2(−1+3x−3x2)−4x3(−1+3x−3x2)⇒−1+3x−3x2−6x2+18x3−18x4+4x3−12x4+12x5⇒−1+3x+(−3−6)x2+(18+4)x3+(−18−12)x4+12x5⇒−1+3x−9x2+22x3−30x4+12x5\Rightarrow (1 + 6x^2 - 4x^3)(-1 + 3x - 3x^2) \\[1em] \Rightarrow 1(-1 + 3x - 3x^2) + 6x^2(-1 + 3x - 3x^2) - 4x^3(-1 + 3x - 3x^2) \\[1em] \Rightarrow -1 + 3x - 3x^2 - 6x^2 + 18x^3 - 18x^4 + 4x^3 - 12x^4 + 12x^5 \\[1em] \Rightarrow -1 + 3x + (-3 - 6)x^2 + (18 + 4)x^3 + (-18 - 12)x^4 + 12x^5 \\[1em] \Rightarrow -1 + 3x - 9x^2 + 22x^3 - 30x^4 + 12x^5

Hence, (1+6x2−4x3)(−1+3x−3x2)=−1+3x−9x2+22x3−30x4+12x5\bm{(1 + 6x^2 - 4x^3)(-1 + 3x - 3x^2) = -1 + 3x - 9x^2 + 22x^3 - 30x^4 + 12x^5}.

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