2.4-Algebraic Identities

2.4-Algebraic Identities Important Formulae

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Grade 9 → Math → Polynomials → 2.4-Algebraic Identities

After successful completion of this topic, you should be able to:

  • To be able to use standard algebraic identities involving squares and cubes.

Identity I: \[(x + y)^2 = x^2 + 2xy + y^2\]
Identity II: \[(x - y)^2 = x^2 - 2xy + y^2\]
Identity III: \[ x^2 - y^2 = (x + y) (x -y) \]
Identity IV: \[ (x + a) (x + b) = x^2 + (a+b)x + ab \]
Identity V: \[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2 xz \]
Identity VI: \[ (x + y)^3 = x^3 + y^3 + 3xy (x + y)\]
Identity VII: \begin{align*} (x - y)^3 &= x^3 - y^3 - 3xy (x - y)\\ &= x^3 - 3x^2y + 3xy^2 - y^3 \end{align*}
Identity VIII: \[ x^3 + y^3 + z^3 - 3xyz = (x + y + z) (x^2 + y^2 + z^2 - xy - yz - zx)\]


Kdkeller, Public domain, via Wikimedia Commons

Use suitable identities to find the following products:
(i) (x+4)(x+10)
(ii) (x+8)(x–10)
(iii) (3x+4)(3x–5)
(iv) $\left(y^2 + \dfrac{3}{2}\right) \left(y^2 – \dfrac{3}{2}\right)$
(v) (3–2x)(3+2x)

Solution:

Finding Products Using Identities

(i) $(x + 4)(x + 10) = x^2 + 14x + 40$.
(ii) $(x + 8)(x - 10) = x^2 - 2x - 80$.
(iii) $(3x + 4)(3x - 5) = 9x^2 - 15x + 12x - 20 = 9x^2 - 3x - 20$.
(iv) $(y^2 + \frac{3}{2})(y^2 - \frac{3}{2}) = y^4 - \left(\frac{3}{2}\right)^2 = y^4 - \frac{9}{4}$.
(v) $(3 - 2x)(3 + 2x) = 9 - (2x)^2 = 9 - 4x^2$.

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Evaluate the following products without multiplying directly:
(i) 103 $\times$ 107
(ii) 95 $\times$ 96
(iii) 104 $\times$ 96

Solution:

Evaluating Products Using Identities

(i) \begin{align*} 103 \times 107 &= (100 + 3)(100 + 7)\\ &= 100^2 + (3 + 7) \times 100 + (3 \times 7)\\ &= 10000 + 1000 + 21\\ &= 11021 \end{align*}
(ii) \begin{align*} 95 \times 96 &= (100 - 5)(100 - 4)\\ &= 100^2 - (5 + 4) \times 100 + (5 \times 4)\\ &= 10000 - 900 + 20\\ &= 9120 \end{align*}
(iii) \begin{align*} 104 \times 96 &= (100 + 4)(100 - 4)\\ &= 100^2 - 4^2\\ &= 10000 - 16\\ &= 9984 \end{align*}

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Factorise the following using appropriate identities:
(i) $9x^2 +6xy+y^2$
(ii) $4y^2 –4y+1$
(iii) $x^2 – 100$

Solution:

Factorisation using Identities

(i) $9x^2 + 6xy + y^2 = (3x + y)^2$
(ii) $4y^2 - 4y + 1 = (2y - 1)^2$
(iii) $x^2 - 100 = (x - 10)(x + 10)$

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Expand each of the following, using suitable identities:

(i) (x+2y+4z)$^2$
(ii) (2x–y+z)$^2$
(iii) (–2x+3y+2z)$^2$
(iv)(3a–7b–c)$^2$
(v)(–2x+5y–3z)$^2$
(vi)$\left(\dfrac{1}{4}a−\dfrac{1}{2}b+1\right)^2$

Solution:

Expansion using Identities

(i) $(x + 2y + 4z)^2 = x^2 + 4y^2 + 16z^2 + 4xy + 8xz + 8yz$
(ii) $(2x - y + z)^2 = 4x^2 + y^2 + z^2 - 4xy + 4xz - 2yz$
(iii) $(-2x + 3y + 2z)^2 = 4x^2 + 9y^2 + 4z^2 - 12xy - 8xz + 12yz$
(iv) $(3a - 7b - c)^2 = 9a^2 + 49b^2 + c^2 - 42ab - 6ac + 21bc$
(v) $(-2x + 5y - 3z)^2 = 4x^2 + 25y^2 + 9z^2 - 20xy + 12xz - 30yz$
(vi) $\left(\dfrac{1}{4}a - \dfrac{1}{2}b + 1\right)^2 = \dfrac{1}{16}a^2 + \dfrac{1}{4}b^2 + 1 - \dfrac{1}{4}ab + \dfrac{1}{2}a - \dfrac{1}{2}b$

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Factorise:

(i) $4x^2 +9y^2 +16z^2 +12xy–24yz–16xz$
(ii) $2x^2+y^2 +8z^2 – 2\sqrt{2} xy+ 4\sqrt{2} yz–8xz$

Solution:

Factorisation

(i) $4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz = (2x + 3y - 4z)^2$
(ii) $2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8xz = (\sqrt{2}x - y + 2\sqrt{2}z)^2$

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Write the following cubes in expanded form:

(i) $(2x+1)^3 $
(ii)$ (2a–3b)^3$

Solution:

Expansion of Cubes

(i) $(2x + 1)^3 = 8x^3 + 12x^2 + 6x + 1$
(ii) $(2a - 3b)^3 = 8a^3 - 36a^2b + 54ab^2 - 27b^3$

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Evaluate the following using suitable identities:

(i) $(99)^3$
(ii) $(102)^3$
(iii) $(998)^3$

Solution:

Evaluate the following using suitable identities

(i) \begin{align*} 99^3 &= (100 - 1)^3 = 100^3 - 3 \cdot 100^2 \cdot 1 + 3 \cdot 100 \cdot 1^2 - 1^3 \\ &= 1000000 - 30000 + 300 - 1 \\ &= 970299 \end{align*}

(ii) \begin{align*} 102^3 &= (100 + 2)^3 = 100^3 + 3 \cdot 100^2 \cdot 2 + 3 \cdot 100 \cdot 2^2 + 2^3 \\ &= 1000000 + 60000 + 1200 + 8 \\ &= 1061208 \end{align*}

(iii) \begin{align*} 998^3 &= (1000 - 2)^3 = 1000^3 - 3 \cdot 1000^2 \cdot 2 + 3 \cdot 1000 \cdot 2^2 - 2^3 \\ &= 1000000000 - 6000000 + 12000 - 8 \\ &= 994011992 \end{align*}

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Factorise each of the following:

(i) $8a^3 + b^3 + 12a^2b + 6ab^2$
(ii) $8a^3 - b^3 – 12a^2b + 6ab^2$
(iii) $27 - 125a^3 - 135a + 225a^2$
(iv)$64a^3 - 27b^3 - 144a^2b + 108ab^2$
(v) $27p^3 - \dfrac{1}{216} - \dfrac{9}{2}p^2 + \dfrac{1}{4}p$

Verify:

(i) $x^3 + y^3 = (x + y)(x^2 – xy + y^2)$
(ii) $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

Factorise each of the following:

(i) $27y^3 + 125z^3$
(ii) $64m^3 - 343n^3$
[Hint : See Question 9.]

Factorise: \[27x^3 + y^3 + z^3 - 9xyz\]

Verify that: \[x^3 + y^3 + z^3 - 3xyz= \dfrac{1}{2}(x+y+z)[(x−y)^2 +(y−z)^2 +(z−x)^2]\]

If x + y + z = 0, show that $x^3 + y^3 + z^3 =3xyz$.

Without actually calculating the cubes, find the value of each of the following:
(i) $(–12)^3 + (7)^3 + (5)^3$
(ii) $(28)^3 + (–15)^3 + (–13)^3$

Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
Area: $25a^2 – 35a + 12$
Area: $35y^2 + 13y – 12$

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
Volume: $3x^2 - 12x$
Volume: $12ky^2 + 8ky - 20k$