7th Class Mathematics Algebraic Expressions Algebraic identities

Algebraic identities

Category : 7th Class

Algebraic identities

Abatement of equality which holds, for all values of the variable is called algebraic identities.

Now we recall some important algebraic identities:

• ${{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$
• ${{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
• ${{a}^{2}}-{{b}^{2}}=(a-b)(a+b)$
• ${{(a+b+c)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca$
• ${{a}^{3}}+{{b}^{3}}={{(a+b)}^{3}}-3ab(a+b)$or ${{(a+b)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab(a+b)$
• ${{(a-b)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab(a-b)$
• ${{a}^{3}}+{{b}^{3}}={{(a+b)}^{3}}-3ab(a+b)$or $(a+b)({{a}^{2}}-ab+{{b}^{2}})$
• ${{a}^{3}}-{{b}^{3}}={{(a-b)}^{3}}+3ab(a-b)$or $(a-b)({{a}^{2}}+ab+{{b}^{2}})$
• ${{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=(a+b+c)$$({{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca)$

If $(a+b+c)=0$${{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc$

Simplify:  ${{(2p+3q-+4r)}^{2}}+{{(2p-3q-4r)}^{2}}$

(a) $2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)$

(b) $-2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)$

(c) $2(-4{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}+16rp)$

(d)$~2(5{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}-98rp)$

(e) None of these

Explanation

Let us first solve,

${{\left[ 2p+3q+(-4r) \right]}^{2}}={{(2p)}^{2}}+{{(3q)}^{2}}$$+{{(-4r)}^{2}}+2(2p)(3q)+2(3q)(-4r)$ $+2(-4r)(2p)=4{{p}^{2}}+4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}$$+12pq-24qr-16rp..........(i)$

Now solve, $~{{(2p-3q-4r)}^{2}}={{\left[ 2p+(-3q)+(-4r) \right]}^{2}}$

$={{(2p)}^{2}}+{{(-3q)}^{2}}+{{(-4r)}^{2}}+2(2p)(-3q)+2(-3q)$$(-4r)+2(-4r)(2p)$

$=4{{p}^{2}}+9{{q}^{2}}+16r2-12pq+24qr-16rp\text{ }.......\left( ii \right)$

Adding, (i) & (ii) we get,

${{(2p+3a-4r)}^{2}}+{{(2p-3q-4r)}^{2}}$

$=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp$+ $(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-12pq+24qr-16rp)$

$=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp+4{{p}^{2}}+9{{q}^{2}}$    $+16{{r}^{2}}-12pq+24qr-16rp$

$=8{{p}^{2}}+18{{q}^{2}}+32{{r}^{2}}-32rp$

$=2(4{{p}^{2}}+9{{q}^{2+}}16r2-16rp)$

The expanded form of ${{(2x+3y-5z)}^{2}}$ is:

(a) $4{{x}^{2}}+9{{y}^{2}}+25{{z}^{2}}+12xy-30yz-20zx$

(b) $5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36{{y}^{6}}+21x{{y}^{2}}$

(c) $8{{a}^{2}}+{{a}^{3}}-{{c}^{2}}+7{{x}^{2}}+2{{c}^{2}}+9{{c}^{2}}y$

(d) $1{{z}^{2}}-{{z}^{4}}-{{8}^{c}}-{{75}^{2}}+2{{c}^{2}}-98{{c}^{2}}$

(e) None of these

Explanation

${{(2x+3y-5z)}^{2}}={{(2x)}^{2}}+{{(3y)}^{2}}+{{(-5z)}^{2}}$$+2(2x)(3y)+2(3y)(-5z)+2(5z)(2x)\text{ }$

Expand: ${{(3a-b+4c)}^{2}}$

(a) $~5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36y+21xy$

(b) $9{{a}^{2}}+{{b}^{2}}+16{{c}^{2}}-6ab-8bc+24ac$

(c) $2{{x}^{2}}+7{{y}^{2}}+2{{z}^{2}}+xy+3yz+2zx$

(d) $15a2+22{{b}^{2}}+2{{c}^{4}}+ab+4bc+102ca$

(e) $5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36y+21xy$

Explanation

${{(3a-b+4c)}^{2}}={{(3a(-b)+4c)}^{2}}={{(3a)}^{2}}$$+{{(-b)}^{2}}+{{(4c)}^{2}}+2(3a)(-b)+2(-b)$ $(4c)+2(4c)(3a)=9{{a}^{2}}+{{b}^{2}}+16{{c}^{2}}-6ab-8bc+24ac$

Find the value of ${{(3a+5b)}^{3}}.$

(a) $27{{a}^{3}}+125{{b}^{3}}+135{{a}^{2}}b+225a{{b}^{2}}$

(b) $27{{a}^{3}}-125{{b}^{3}}-135{{a}^{2}}b+2225a{{b}^{2}}$

(c) $27{{a}^{2}}+155{{b}^{2}}-135{{a}^{2}}b-225{{a}^{2}}b$

(d) $29{{a}^{2}}-156{{b}^{2}}-156{{a}^{2}}b-225{{a}^{4}}c$

(e) None of these

Find the Cube of  $x-2y.$

(a)$~{{x}^{3}}+8{{y}^{2}}+6{{x}^{2}}y-12x{{y}^{3}}$

(b) ${{x}^{3}}-8{{y}^{3}}-6{{x}^{2}}y+12x{{y}^{2}}$v

(c)${{x}^{2}}+87y+7xy-7xy$

(d) $7x{{y}^{3}}-6x-74{{x}^{2}}y+2xy$

(e) None of these

Evaluate: ${{\left( -6p+\frac{1}{3}q-r \right)}^{2}}$

(a) $36{{p}^{2}}\frac{9}{1}{{p}^{2}}+{{5}^{2}}-4pq-\frac{2}{3}gr+12rp$

(b) $36{{p}^{3}}\frac{9}{1}{{q}^{2}}+{{p}^{2}}+4qp-\frac{3}{2}qr+13rp$

(c)$~36{{p}^{2}}+\frac{1}{9}{{q}^{2}}+{{r}^{2}}-4qp-\frac{2}{3}qr+12rp$

(d) $36{{x}^{2}}-\frac{99}{8}{{s}^{2}}-{{s}^{2}}-4pq-\frac{2}{3}qr+14rp$

(e) None of these