Algebraic Expressions and Identities — Check Your Progress

Add the following expressions:

$-5x^2y + 3xy^2 - 7xy + 8, 12x^2y - 5xy^2 + 3xy - 2$

Answer

Write each expression in a separate row, keeping the like terms one below the other, and then add them.

$\begin{array}{rrrrr} & - 5x^2y & + 3xy^2 & - 7xy & + 8 \\ + & 12x^2y & - 5xy^2 & + 3xy & - 2 \\ \hline & 7x^2y & - 2xy^2 & - 4xy & + 6 \\ \end{array}$

Hence, the required sum = $\bm{7x^2y - 2xy^2 - 4xy + 6}$.

Add the following expressions:

$9xy + 3yz - 5zx, 4yz + 9zx - 5y, -5xz + 2x - 5xy$

Answer

As $zx$ is the same as $xz$, the third expression can be written as $-5zx + 2x - 5xy$.

Write each expression in a separate row, keeping the like terms one below the other, and then add them.

$\begin{array}{rrrrrr} & 9xy & + 3yz & - 5zx & & \\ & & + 4yz & + 9zx & & - 5y \\ + & - 5xy & & - 5zx & + 2x & \\ \hline & 4xy & + 7yz & - zx & + 2x & - 5y \\ \end{array}$

Hence, the required sum = $\bm{4xy + 7yz - zx + 2x - 5y}$.

Subtract:

$5a - 3b + 11c - 2$ from $3a + 5b - 9c + 3$

Answer

Subtracting,

$\begin{array}{rrrrr} & 3a & + 5b & - 9c & + 3 \\ & 5a & - 3b & + 11c & - 2 \\ &- &+ &- &+ \\ \hline & - 2a & + 8b & - 20c & + 5 \\ \end{array}$

Hence, the required difference = $\bm{-2a + 8b - 20c + 5}$.

Subtract:

$10x^2 - 8y^2 + 5y - 3$ from $8x^2 - 5xy + 2y^2 + 5x - 3y$

Answer

Subtracting,

$\begin{array}{rrrrrrr} & 8x^2 & - 5xy & + 2y^2 & + 5x & - 3y & \\ & 10x^2 & & - 8y^2 & & + 5y & - 3 \\ &- & &+ & &- &+ \\ \hline & - 2x^2 & - 5xy & + 10y^2 & + 5x & - 8y & + 3 \\ \end{array}$

Hence, the required difference = $\bm{-2x^2 - 5xy + 10y^2 + 5x - 8y + 3}$.

What must be added to $5x^2 - 3x + 1$ to get $3x^3 - 7x^2 + 8$?

Answer

Let the required expression be A.

According to the question,

$(5x^2 - 3x + 1) + A = 3x^3 - 7x^2 + 8$

⇒ $A = (3x^3 - 7x^2 + 8) - (5x^2 - 3x + 1)$

Subtracting,

$\begin{array}{rrrrr} & 3x^3 & - 7x^2 & & + 8 \\ & & 5x^2 & - 3x & + 1 \\ & &- &+ &- \\ \hline & 3x^3 & - 12x^2 & + 3x & + 7 \\ \end{array}$

Hence, $\bm{3x^3 - 12x^2 + 3x + 7}$ must be added.

Find the product of

$3x^2y$ and $-4xy^2$

Answer

Solving,

$\Rightarrow 3x^2y \times (-4xy^2)\\[1em] \Rightarrow (3 \times (-4)) \times (x^2 \times x) \times (y \times y^2)\\[1em] \Rightarrow -12x^{(2+1)}y^{(1+2)}\\[1em] \Rightarrow -12x^3y^3$

Hence, the required product = $\bm{-12x^3y^3}$.

Find the product of

$-\dfrac{4}{5}xy, \dfrac{5}{7}yz$ and $-\dfrac{14}{9}zx$

Answer

Solving,

$\Rightarrow -\dfrac{4}{5}xy \times \dfrac{5}{7}yz \times \Big(-\dfrac{14}{9}zx\Big)\\[1em] \Rightarrow \Big(-\dfrac{4}{5} \times \dfrac{5}{7} \times -\dfrac{14}{9}\Big) \times (x \times x) \times (y \times y) \times (z \times z)\\[1em] \Rightarrow \dfrac{4 \times 5 \times 14}{5 \times 7 \times 9}x^2y^2z^2\\[1em] \Rightarrow \dfrac{280}{315}x^2y^2z^2\\[1em] \Rightarrow \dfrac{8}{9}x^2y^2z^2$

Hence, the required product = $\bm{\dfrac{8}{9}x^2y^2z^2}$.

Multiply:

$(3pq - 4p^2 + 5q^2 + 7)$ by $-7pq$

Answer

Solving,

$\Rightarrow (3pq - 4p^2 + 5q^2 + 7) \times (-7pq)\\[1em] \Rightarrow 3pq \times (-7pq) - 4p^2 \times (-7pq) + 5q^2 \times (-7pq) + 7 \times (-7pq)\\[1em] \Rightarrow -21p^2q^2 + 28p^3q - 35pq^3 - 49pq$

Hence, the required product = $\bm{-21p^2q^2 + 28p^3q - 35pq^3 - 49pq}$.

Multiply:

$\left(\dfrac{3}{4}x^2y - \dfrac{4}{5}xy + \dfrac{5}{6}xy^2\right)$ by $-15xyz$

Answer

Solving,

$\Rightarrow \Big(\dfrac{3}{4}x^2y - \dfrac{4}{5}xy + \dfrac{5}{6}xy^2\Big) \times (-15xyz)\\[1em] \Rightarrow \dfrac{3}{4}x^2y \times (-15xyz) - \dfrac{4}{5}xy \times (-15xyz) + \dfrac{5}{6}xy^2 \times (-15xyz)\\[1em] \Rightarrow -\dfrac{3 \times 15}{4}x^3y^2z + \dfrac{4 \times 15}{5}x^2y^2z - \dfrac{5 \times 15}{6}x^2y^3z\\[1em] \Rightarrow -\dfrac{45}{4}x^3y^2z + 12x^2y^2z - \dfrac{25}{2}x^2y^3z$

Hence, the required product = $\bm{-\dfrac{45}{4}x^3y^2z + 12x^2y^2z - \dfrac{25}{2}x^2y^3z}$.

Multiply:

$(5x^2 + 4x - 2)$ by $(3 - x - 4x^2)$

Answer

$(5x^2 + 4x - 2)(3 - x - 4x^2)$

Arranging the terms of the second polynomial in descending powers of $x$,

$\Rightarrow (5x^2 + 4x - 2)(-4x^2 - x + 3)\\[1em] \Rightarrow 5x^2(-4x^2 - x + 3) + 4x(-4x^2 - x + 3) - 2(-4x^2 - x + 3)\\[1em] \Rightarrow -20x^4 - 5x^3 + 15x^2 - 16x^3 - 4x^2 + 12x + 8x^2 + 2x - 6\\[1em] \Rightarrow -20x^4 + (-5x^3 - 16x^3) + (15x^2 - 4x^2 + 8x^2) + (12x + 2x) - 6\\[1em] \Rightarrow -20x^4 - 21x^3 + 19x^2 + 14x - 6$

Hence, the required product = $\bm{-20x^4 - 21x^3 + 19x^2 + 14x - 6}$.

Multiply:

$(7x^2 + 12xy - 9y^2)$ by $(3x^2 - 5xy + 3y^2)$

Answer

Solving,

$\Rightarrow (7x^2 + 12xy - 9y^2)(3x^2 - 5xy + 3y^2)\\[1em] \Rightarrow 7x^2(3x^2 - 5xy + 3y^2) + 12xy(3x^2 - 5xy + 3y^2) - 9y^2(3x^2 - 5xy + 3y^2)\\[1em] \Rightarrow 21x^4 - 35x^3y + 21x^2y^2 + 36x^3y - 60x^2y^2 + 36xy^3 - 27x^2y^2 + 45xy^3 - 27y^4\\[1em] \Rightarrow 21x^4 + (-35x^3y + 36x^3y) + (21x^2y^2 - 60x^2y^2 - 27x^2y^2) + (36xy^3 + 45xy^3) - 27y^4\\[1em] \Rightarrow 21x^4 + x^3y - 66x^2y^2 + 81xy^3 - 27y^4$

Hence, the required product = $\bm{21x^4 + x^3y - 66x^2y^2 + 81xy^3 - 27y^4}$.

Simplify the following expression and evaluate it as directed:

$(3ab - 2a^2 + 5b^2) \times (2b^2 - 5ab + 3a^2) + 8a^3b - 7b^4$ for $a = 1, b = -1$

Answer

$(3ab - 2a^2 + 5b^2) \times (2b^2 - 5ab + 3a^2) + 8a^3b - 7b^4$

First we find the product of the two polynomials.

$\Rightarrow (3ab - 2a^2 + 5b^2)(2b^2 - 5ab + 3a^2)\\[1em] \Rightarrow 3ab(2b^2 - 5ab + 3a^2) - 2a^2(2b^2 - 5ab + 3a^2) + 5b^2(2b^2 - 5ab + 3a^2)\\[1em] \Rightarrow 6ab^3 - 15a^2b^2 + 9a^3b - 4a^2b^2 + 10a^3b - 6a^4 + 10b^4 - 25ab^3 + 15a^2b^2\\[1em] \Rightarrow -6a^4 + (9a^3b + 10a^3b) + (-15a^2b^2 - 4a^2b^2 + 15a^2b^2) + (6ab^3 - 25ab^3) + 10b^4\\[1em] \Rightarrow -6a^4 + 19a^3b - 4a^2b^2 - 19ab^3 + 10b^4$

Now, adding $8a^3b - 7b^4$,

$\Rightarrow -6a^4 + 19a^3b - 4a^2b^2 - 19ab^3 + 10b^4 + 8a^3b - 7b^4\\[1em] \Rightarrow -6a^4 + (19a^3b + 8a^3b) - 4a^2b^2 - 19ab^3 + (10b^4 - 7b^4)\\[1em] \Rightarrow 27a^3b - 4a^2b^2 - 19ab^3 - 6a^4 + 3b^4$

For $a = 1, b = -1$,

$\Rightarrow 27(1)^3(-1) - 4(1)^2(-1)^2 - 19(1)(-1)^3 - 6(1)^4 + 3(-1)^4\\[1em] \Rightarrow -27 - 4 + 19 - 6 + 3\\[1em] \Rightarrow -15$

Hence, the simplified expression = $\bm{27a^3b - 4a^2b^2 - 19ab^3 - 6a^4 + 3b^4}$ and its value = -15.

Simplify the following expression and evaluate it as directed:

$(1.7x - 2.5y)(2y + 3x + 4) - 7.8x^2 - 10y$ for $x = 0, y = 1$

Answer

Solving,

$\Rightarrow (1.7x - 2.5y)(2y + 3x + 4) - 7.8x^2 - 10y\\[1em] \Rightarrow 1.7x(2y + 3x + 4) - 2.5y(2y + 3x + 4) - 7.8x^2 - 10y\\[1em] \Rightarrow 3.4xy + 5.1x^2 + 6.8x - 5y^2 - 7.5xy - 10y - 7.8x^2 - 10y\\[1em] \Rightarrow (5.1x^2 - 7.8x^2) + (3.4xy - 7.5xy) - 5y^2 + 6.8x + (-10y - 10y)\\[1em] \Rightarrow -2.7x^2 - 4.1xy - 5y^2 + 6.8x - 20y$

For $x = 0, y = 1$,

$\Rightarrow -2.7(0)^2 - 4.1(0)(1) - 5(1)^2 + 6.8(0) - 20(1)\\[1em] \Rightarrow 0 - 0 - 5 + 0 - 20\\[1em] \Rightarrow -25$

Hence, the simplified expression = $\bm{-2.7x^2 - 4.1xy - 5y^2 + 6.8x - 20y}$ and its value = -25.

Carry out the following division:

$66pq^2r^3 \div 11qr^2$

Answer

Solving,

$\Rightarrow 66pq^2r^3 \div 11qr^2\\[1em] \Rightarrow \dfrac{66pq^2r^3}{11qr^2}\\[1em] \Rightarrow \dfrac{66}{11} \times p \times q^{2-1} \times r^{3-2}\\[1em] \Rightarrow 6pqr$

Hence, the required quotient = $\bm{6pqr}$.

Carry out the following division:

$(x^3 + 2x^2 + 3x) \div 2x$

Answer

Solving,

$\Rightarrow (x^3 + 2x^2 + 3x) \div 2x\\[1em] \Rightarrow \dfrac{x^3}{2x} + \dfrac{2x^2}{2x} + \dfrac{3x}{2x}\\[1em] \Rightarrow \dfrac{x^{3-1}}{2} + x^{2-1} + \dfrac{3}{2}\\[1em] \Rightarrow \dfrac{x^2}{2} + x + \dfrac{3}{2}$

Hence, the required quotient = $\bm{\dfrac{x^2}{2} + x + \dfrac{3}{2}}$.

Divide $10x^4 - 19x^3 + 17x^2 + 15x - 42$ by $2x^2 - 3x + 5$

Answer

Dividing $10x^4 - 19x^3 + 17x^2 + 15x - 42$ by $2x^2 - 3x + 5$

$\begin{array}{l} \phantom{2x^2 - 3x + 5\big)}{5x^2 - 2x - 7} \\ 2x^2 - 3x + 5\overline{\smash{\big)}10x^4 - 19x^3 + 17x^2 + 15x - 42} \\ \phantom{+++++()}\underline{\underset{-}{}10x^4 \underset{+}{-} 15x^3 \underset{-}{+} 25x^2 \phantom{+ 15x} \phantom{- 42}} \\ \phantom{+++++++++}-4x^3 - 8x^2 + 15x - 42 \\ \phantom{+++++++++()}\underline{\underset{+}{-}4x^3 \underset{-}{+} 6x^2 \underset{+}{-} 10x \phantom{- 42}} \\ \phantom{++++++++++++}-14x^2 + 25x - 42 \\ \phantom{++++++++++++}\underline{\underset{+}{-}14x^2 \underset{-}{+} 21x \underset{+}{-} 35} \\ \phantom{2x^2 - 3x + 5\big)10x^4 - 19x^3 + 17x^2 + 15x;}4x - 7 \end{array}$

Hence, quotient = $\bm{5x^2 - 2x - 7}$ and remainder = $\bm{4x - 7}$.

Using identities, find the following product:

$(3x + 4y)(3x + 4y)$

Answer

$\Rightarrow (3x + 4y)(3x + 4y) = (3x + 4y)^2$

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$,

$\Rightarrow (3x)^2 + 2 \times 3x \times 4y + (4y)^2\\[1em] \Rightarrow 9x^2 + 24xy + 16y^2$

Hence, $\bm{(3x + 4y)(3x + 4y) = 9x^2 + 24xy + 16y^2}$.

Using identities, find the following product:

$\left(\dfrac{5}{2}a - b\right)\left(\dfrac{5}{2}a - b\right)$

Answer

Using the identity $(a - b)^2 = a^2 - 2ab + b^2$,

$\Rightarrow \Big(\dfrac{5}{2}a - b\Big)\Big(\dfrac{5}{2}a - b\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{2}a - b\Big)^2\\[1em] \Rightarrow \Big(\dfrac{5}{2}a\Big)^2 - 2 \times \dfrac{5}{2}a \times b + (b)^2\\[1em] \Rightarrow \dfrac{25}{4}a^2 - 5ab + b^2$

Hence, $\bm{\Big(\dfrac{5}{2}a - b\Big)\Big(\dfrac{5}{2}a - b\Big) = \dfrac{25}{4}a^2 - 5ab + b^2}$.

Using identities, find the following product:

$(3.5m - 1.5n)(3.5m + 1.5n)$

Answer

$\Rightarrow (3.5m - 1.5n)(3.5m + 1.5n)$

Using the identity $(a - b)(a + b) = a^2 - b^2$,

$\Rightarrow (3.5m)^2 - (1.5n)^2\\[1em] \Rightarrow 12.25m^2 - 2.25n^2$

Hence, $\bm{(3.5m - 1.5n)(3.5m + 1.5n) = 12.25m^2 - 2.25n^2}$.

Using identities, find the following product:

$(7xy - 2)(7xy + 7)$

Answer

$\Rightarrow (7xy - 2)(7xy + 7) = (7xy + (-2))(7xy + 7)$

Using the identity $(x + a)(x + b) = x^2 + (a + b)x + ab$,

$\Rightarrow (7xy)^2 + ((-2) + 7)(7xy) + (-2) \times 7\\[1em] \Rightarrow 49x^2y^2 + 5 \times 7xy - 14\\[1em] \Rightarrow 49x^2y^2 + 35xy - 14$

Hence, $\bm{(7xy - 2)(7xy + 7) = 49x^2y^2 + 35xy - 14}$.

Using suitable identities, evaluate the following:

105²

Answer

⇒ 105² = (100 + 5)²

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$,

⇒ (100)² + 2 × 100 × 5 + (5)²

⇒ 10000 + 1000 + 25

⇒ 11025

Hence, 105² = 11025.

Using suitable identities, evaluate the following:

97²

Answer

⇒ 97² = (100 - 3)²

Using the identity $(a - b)^2 = a^2 - 2ab + b^2$,

⇒ (100)² - 2 × 100 × 3 + (3)²

⇒ 10000 - 600 + 9

⇒ 9409

Hence, 97² = 9409.

Using suitable identities, evaluate the following:

201 × 199

Answer

⇒ 201 × 199 = (200 + 1)(200 - 1)

Using the identity $(a + b)(a - b) = a^2 - b^2$,

⇒ (200)² - (1)²

⇒ 40000 - 1

⇒ 39999

Hence, 201 × 199 = 39999.

Using suitable identities, evaluate the following:

87² - 13²

Answer

⇒ 87² - 13²

Using the identity $a^2 - b^2 = (a + b)(a - b)$,

⇒ (87 + 13)(87 - 13)

⇒ 100 × 74

⇒ 7400

Hence, 87² - 13² = 7400.

Using suitable identities, evaluate the following:

105 × 107

Answer

⇒ 105 × 107 = (100 + 5)(100 + 7)

Using the identity $(x + a)(x + b) = x^2 + (a + b)x + ab$,

⇒ (100)² + (5 + 7) × 100 + 5 × 7

⇒ 10000 + 1200 + 35

⇒ 11235

Hence, 105 × 107 = 11235.

Prove the following:

$(a + b)^2 = (a - b)^2 + 4ab$

Answer

Taking R.H.S. = $(a - b)^2 + 4ab$

Using the identity $(a - b)^2 = a^2 - 2ab + b^2$,

$\Rightarrow a^2 - 2ab + b^2 + 4ab\\[1em] \Rightarrow a^2 + 2ab + b^2\\[1em] \Rightarrow (a + b)^2$

Since, L.H.S. = R.H.S.

Hence, proved that $\bm{(a + b)^2 = (a - b)^2 + 4ab}$.

Prove the following:

$(2a + 3b)^2 + (2a - 3b)^2 = 8a^2 + 18b^2$

Answer

Taking L.H.S. = $(2a + 3b)^2 + (2a - 3b)^2$

Using the identities $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$,

$\Rightarrow [(2a)^2 + 2 \times 2a \times 3b + (3b)^2] + [(2a)^2 - 2 \times 2a \times 3b + (3b)^2]\\[1em] \Rightarrow 4a^2 + 12ab + 9b^2 + 4a^2 - 12ab + 9b^2\\[1em] \Rightarrow (4a^2 + 4a^2) + (12ab - 12ab) + (9b^2 + 9b^2)\\[1em] \Rightarrow 8a^2 + 18b^2$

Since, L.H.S. = R.H.S.

Hence, proved that $\bm{(2a + 3b)^2 + (2a - 3b)^2 = 8a^2 + 18b^2}$.

If $x + \dfrac{1}{x} = 5$, evaluate

(i) $x^2 + \dfrac{1}{x^2}$

(ii) $x^4 + \dfrac{1}{x^4}$

Answer

(i) Given,

$x + \dfrac{1}{x} = 5$

Squaring both sides,

$\Rightarrow \Big(x + \dfrac{1}{x}\Big)^2 = 5^2\\[1em] \Rightarrow x^2 + 2 \times x \times \dfrac{1}{x} + \dfrac{1}{x^2} = 25\\[1em] \Rightarrow x^2 + 2 + \dfrac{1}{x^2} = 25\\[1em] \Rightarrow x^2 + \dfrac{1}{x^2} = 25 - 2\\[1em] \Rightarrow x^2 + \dfrac{1}{x^2} = 23$

Hence, $\bm{x^2 + \dfrac{1}{x^2} = 23}$.

(ii) From part (i), $x^2 + \dfrac{1}{x^2} = 23$

Squaring both sides,

$\Rightarrow \Big(x^2 + \dfrac{1}{x^2}\Big)^2 = 23^2\\[1em] \Rightarrow (x^2)^2 + 2 \times x^2 \times \dfrac{1}{x^2} + \Big(\dfrac{1}{x^2}\Big)^2 = 529\\[1em] \Rightarrow x^4 + 2 + \dfrac{1}{x^4} = 529\\[1em] \Rightarrow x^4 + \dfrac{1}{x^4} = 529 - 2\\[1em] \Rightarrow x^4 + \dfrac{1}{x^4} = 527$

Hence, $\bm{x^4 + \dfrac{1}{x^4} = 527}$.

If $a + b = 5$ and $a^2 + b^2 = 13$, find $ab$

Answer

Given,

$a + b = 5$ and $a^2 + b^2 = 13$

We know that $(a + b)^2 = a^2 + b^2 + 2ab$

Substituting the values,

$\Rightarrow (5)^2 = 13 + 2ab\\[1em] \Rightarrow 25 = 13 + 2ab\\[1em] \Rightarrow 2ab = 25 - 13\\[1em] \Rightarrow 2ab = 12\\[1em] \Rightarrow ab = 6$

Hence, $\bm{ab = 6}$.

Find the polynomial that represents the product of three consecutive odd integers, the smallest being $(2x - 1)$

Answer

Consecutive odd integers differ by 2.

Given, the smallest odd integer = $2x - 1$

So, the next odd integer = $2x - 1 + 2 = 2x + 1$

And the third odd integer = $2x + 1 + 2 = 2x + 3$

Required polynomial = $(2x - 1)(2x + 1)(2x + 3)$

= $[(2x - 1)(2x + 1)](2x + 3)$

Using the identity $(a - b)(a + b) = a^2 - b^2$,

$\Rightarrow [(2x)^2 - (1)^2](2x + 3)\\[1em] \Rightarrow (4x^2 - 1)(2x + 3)\\[1em] \Rightarrow 4x^2(2x + 3) - 1(2x + 3)\\[1em] \Rightarrow 8x^3 + 12x^2 - 2x - 3$

Hence, the required polynomial = $\bm{8x^3 + 12x^2 - 2x - 3}$.

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, derive the formula for $(a + b + c)^2$. Hence, find the value of $(2x - 3y + 4z)^2$

Answer

$(a + b + c)^2 = [(a + b) + c]^2$

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, taking $(a + b)$ as the first term and $c$ as the second term,

$\Rightarrow (a + b)^2 + 2(a + b)c + c^2\\[1em] \Rightarrow a^2 + 2ab + b^2 + 2ac + 2bc + c^2\\[1em] \Rightarrow a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Hence, $\bm{(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca}$.

Now, for $(2x - 3y + 4z)^2$, we take $a = 2x, b = -3y$ and $c = 4z$.

$\Rightarrow (2x - 3y + 4z)^2\\[1em] \Rightarrow (2x)^2 + (-3y)^2 + (4z)^2 + 2(2x)(-3y) + 2(-3y)(4z) + 2(4z)(2x)\\[1em] \Rightarrow 4x^2 + 9y^2 + 16z^2 - 12xy - 24yz + 16zx$

Hence, $\bm{(2x - 3y + 4z)^2 = 4x^2 + 9y^2 + 16z^2 - 12xy - 24yz + 16zx}$.

Using the product of algebraic expressions, find the formulas for $(a + b)^3$ and $(a - b)^3$.

Answer

Formula for $\bm{(a + b)^3}$:

$(a + b)^3 = (a + b)(a + b)^2$

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$,

$\Rightarrow (a + b)(a^2 + 2ab + b^2)\\[1em] \Rightarrow a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2)\\[1em] \Rightarrow a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3\\[1em] \Rightarrow a^3 + (2a^2b + a^2b) + (ab^2 + 2ab^2) + b^3\\[1em] \Rightarrow a^3 + 3a^2b + 3ab^2 + b^3\\[1em] \Rightarrow a^3 + b^3 + 3ab(a + b)$

Formula for $\bm{(a - b)^3}$:

$(a - b)^3 = (a - b)(a - b)^2$

Using the identity $(a - b)^2 = a^2 - 2ab + b^2$,

$\Rightarrow (a - b)(a^2 - 2ab + b^2)\\[1em] \Rightarrow a(a^2 - 2ab + b^2) - b(a^2 - 2ab + b^2)\\[1em] \Rightarrow a^3 - 2a^2b + ab^2 - a^2b + 2ab^2 - b^3\\[1em] \Rightarrow a^3 + (-2a^2b - a^2b) + (ab^2 + 2ab^2) - b^3\\[1em] \Rightarrow a^3 - 3a^2b + 3ab^2 - b^3\\[1em] \Rightarrow a^3 - b^3 - 3ab(a - b)$

Hence, $\bm{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a + b)}$ and $\bm{(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a - b)}$.