Algebraic Identities

Q1: Find the following products:

i.  \((x+5)(x+7)\)

Step 1: Use the identity \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=5\) and \(b=7\) ⟹ \(a+b=12\) and \(ab=35\).
Step 3: Substitute in the identity: \(x^2+(12)x+35=x^2+12x+35\).
Answer:\(x^2+12x+35\)

ii.  \((b+2)(b+9)\)

Step 1: Use \((b+m)(b+n)=b^2+(m+n)b+mn\).
Step 2: Here \(m=2\) and \(n=9\) ⟹ \(m+n=11\) and \(mn=18\).
Step 3: Substitute: \(b^2+11b+18\).
Answer:\(b^2+11b+18\)

iii.  \((c+2)\left(c+\frac{3}{5}\right)\)

Step 1: Use \((c+m)(c+n)=c^2+(m+n)c+mn\).
Step 2: Here \(m=2\) and \(n=\frac{3}{5}\) ⟹ \(m+n=2+\frac{3}{5}=\frac{13}{5}\) and \(mn=2\cdot\frac{3}{5}=\frac{6}{5}\).
Step 3: Substitute: \(c^2+\frac{13}{5}c+\frac{6}{5}\).
Answer:\(c^2+\frac{13}{5}c+\frac{6}{5}\)

iv.  \(\left(t+\frac{4}{3}\right)\left(t+\frac{1}{3}\right)\)

Step 1: Use \((t+m)(t+n)=t^2+(m+n)t+mn\).
Step 2: Here \(m=\frac{4}{3}\) and \(n=\frac{1}{3}\) ⟹ \(m+n=\frac{5}{3}\) and \(mn=\frac{4}{9}\).
Step 3: Substitute: \(t^2+\frac{5}{3}t+\frac{4}{9}\).
Answer:\(t^2+\frac{5}{3}t+\frac{4}{9}\)

Q2: Find the following products:

i.  \((y+8)(y-4)\)

Step 1: Use the identity \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=8\) and \(b=-4\)\;⟹\; \(a+b=8+(-4)=4\) and \(ab=8\times(-4)=-32\).
Step 3: Substitute in the identity: \(y^2+(4)y+(-32)=y^2+4y-32\).
Answer:\(y^2+4y-32\)

ii.  \((z+6)(z-11)\)

Step 1: Use \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=6\) and \(b=-11\)\;⟹\; \(a+b=6+(-11)=-5\) and \(ab=6\times(-11)=-66\).
Step 3: Substitute: \(z^2+(-5)z+(-66)=z^2-5z-66\).
Answer:\(z^2-5z-66\)

iii.  \((c-5)(c+1)\)

Step 1: Use \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-5\) and \(b=1\)\;⟹\; \(a+b=-5+1=-4\) and \(ab=(-5)\times1=-5\).
Step 3: Substitute: \(c^2+(-4)c+(-5)=c^2-4c-5\).
Answer:\(c^2-4c-5\)

iv.  \((b-13)(b+10)\)

Step 1: Use \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-13\) and \(b=10\)\;⟹\; \(a+b=-13+10=-3\) and \(ab=(-13)\times10=-130\).
Step 3: Substitute: \(b^2+(-3)b+(-130)=b^2-3b-130\).
Answer:\(b^2-3b-130\)

Q3: Find the following products:

i.  \((x-3)(x-6)\)

Step 1: Use the identity \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-3\) and \(b=-6\)\;⟹\; \(a+b=-3+(-6)=-9\), \(ab=(-3)\times(-6)=18\).
Step 3: Substitute: \(x^2+(-9)x+18=x^2-9x+18\).
Answer:\(x^2-9x+18\)

ii.  \((z-11)(z-4)\)

Step 1: Apply \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-11\), \(b=-4\)\;⟹\; \(a+b=-15\), \(ab=(-11)\times(-4)=44\).
Step 3: Substitute: \(z^2+(-15)z+44=z^2-15z+44\).
Answer:\(z^2-15z+44\)

iii.  \((b-6)(b-8)\)

Step 1: Use \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-6\), \(b=-8\)\;⟹\; \(a+b=-14\), \(ab=(-6)\times(-8)=48\).
Step 3: Substitute: \(b^2+(-14)b+48=b^2-14b+48\).
Answer:\(b^2-14b+48\)

iv.  \(\left(a-\frac{3}{5}\right)\left(a-\frac{1}{3}\right)\)

Step 1: Apply the identity \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-\frac{3}{5}\), \(b=-\frac{1}{3}\).
\(a+b=-\frac{3}{5}-\frac{1}{3}=\frac{-9-5}{15}=\frac{-14}{15}\).
\(ab=\left(-\frac{3}{5}\right)\left(-\frac{1}{3}\right)=\frac{3}{15}=\frac{1}{5}\).
Step 3: Substitute: \(a^2+\left(\frac{-14}{15}\right)a+\frac{1}{5}\).
Answer:\(a^2-\frac{14}{15}a+\frac{1}{5}\)

Q4: Find the following products:

i.  \(\left(x-\frac{1}{2}\right)\left(x+\frac{3}{2}\right)\)

Step 1: Recall the identity \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-\frac{1}{2}\), \(b=\frac{3}{2}\).
\(a+b=-\frac{1}{2}+\frac{3}{2}=\frac{2}{2}=1\).
\(ab=\left(-\frac{1}{2}\right)\left(\frac{3}{2}\right)=-\frac{3}{4}\).
Step 3: Substitute in the identity:
\(x^2+(1)x-\frac{3}{4}=x^2+x-\frac{3}{4}\).
Answer:\(x^2+x-\frac{3}{4}\)

ii.  \(\left(p-3\right)\left(p+\frac{1}{2}\right)\)

Step 1: Use \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-3\), \(b=\frac{1}{2}\).
\(a+b=-3+\frac{1}{2}=\frac{-6+1}{2}=-\frac{5}{2}\).
\(ab=(-3)\times\frac{1}{2}=-\frac{3}{2}\).
Step 3: Substitute in the identity:
\(p^2+(-\frac{5}{2})p-\frac{3}{2}=p^2-\frac{5}{2}p-\frac{3}{2}\).
Answer:\(p^2-\frac{5}{2}p-\frac{3}{2}\)

Q5: Find the following products:

i.  \(\left(4p+3\right)\left(4p+7\right)\)

Step 1: Use identity \((x+a)(x+b)=x^2+(a+b)x+ab\), with \(x=4p\).
Step 2: Here \(a=3\), \(b=7\).
\(a+b=10\), \(ab=21\).
Step 3: Substitute:
\((4p)^2+10(4p)+21=16p^2+40p+21\).
Answer:\(16p^2+40p+21\)

ii.  \(\left(9c+4\right)\left(9c-2\right)\)

Step 1: Take \(x=9c\), apply \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=4\), \(b=-2\).
\(a+b=2\), \(ab=-8\).
Step 3: Substitute:
\((9c)^2+2(9c)-8=81c^2+18c-8\).
Answer:\(81c^2+18c-8\)

iii.  \(\left(3a-8\right)\left(3a+2\right)\)

Step 1: Take \(x=3a\), apply \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-8\), \(b=2\).
\(a+b=-6\), \(ab=-16\).
Step 3: Substitute:
\((3a)^2+(-6)(3a)-16=9a^2-18a-16\).
Answer:\(9a^2-18a-16\)

iv.  \(\left(5x-2\right)\left(5x-7\right)\)

Step 1: Take \(x=5x\), apply \((x+a)(x+b)=x^2+(a+b)x+ab\).
Step 2: Here \(a=-2\), \(b=-7\).
\(a+b=-9\), \(ab=14\).
Step 3: Substitute:
\((5x)^2+(-9)(5x)+14=25x^2-45x+14\).
Answer:\(25x^2-45x+14\)

Q6: Find the following products:

i.  \(\left(x^2+3\right)\left(x^2+6\right)\)

Step 1: Expand using distributive property: \((a+b)(c+d)=ac+ad+bc+bd\).
Step 2: Multiply term by term:
\((x^2)(x^2)+(x^2)(6)+(3)(x^2)+(3)(6)\).
Step 3: Simplify:
\(x^4+6x^2+3x^2+18 = x^4 + 9x^2 + 18\).
Answer:\(x^4 + 9x^2 + 18\)

ii.  \(\left(y^2-1\right)\left(y^2+4\right)\)

Step 1: Expand using distributive law.
\((y^2)(y^2)+(y^2)(4)+(-1)(y^2)+(-1)(4)\).
Step 2: Simplify:
\(y^4+4y^2-y^2-4\).
\(=y^4+3y^2-4\).
Answer:\(y^4+3y^2-4\)

iii.  \(\left(z^2+3\right)\left(z^2-7\right)\)

Step 1: Expand terms:
\((z^2)(z^2)+(z^2)(-7)+(3)(z^2)+(3)(-7)\).
Step 2: Simplify:
\(z^4-7z^2+3z^2-21\).
\(=z^4-4z^2-21\).
Answer:\(z^4-4z^2-21\)

iv.  \(\left(t^2-2\right)\left(t^2-5\right)\)

Step 1: Expand terms:
\((t^2)(t^2)+(t^2)(-5)+(-2)(t^2)+(-2)(-5)\).
Step 2: Simplify:
\(t^4-5t^2-2t^2+10\).
\(=t^4-7t^2+10\).
Answer:\(t^4-7t^2+10\)

Q7: Find the following products:

i. \((ab – 2)(ab + 4)\)

Step 1: Recognize the form \[ (x + p)(x + q) = x^2 + (p+q)x + pq \] Here, \(x = ab\), \(p = -2\), \(q = 4\).
Step 2: Apply identity \[ (ab – 2)(ab + 4) = (ab)^2 + (4 – 2)(ab) + (-2)(4) \]Step 3: Simplify \[ = a^2b^2 + 2ab – 8 \]Answer: \((ab – 2)(ab + 4) = a^2b^2 + 2ab – 8\)

ii. \((2 + xy)(3 – xy)\)

Step 1: Recognize the form \[ (p + q)(r – q) = pr + (r – p)q – q^2 \] Here, \(p = 2\), \(r = 3\), \(q = xy\).
Step 2: Expand using distributive law \[ (2 + xy)(3 – xy) = (2)(3) + (2)(-xy) + (xy)(3) + (xy)(-xy) \]Step 3: Simplify \[ = 6 – 2xy + 3xy – (xy)^2 \\ = 6 + xy – x^2y^2 \]Answer: \((2 + xy)(3 – xy) = 6 + xy – x^2y^2\)

Q8: Find the following products:

i.  \((3x+4y)(4x+3y)\)

Step 1: Recall identity: \((a+b)(c+d) = ac + ad + bc + bd\).
Step 2: Multiply term by term:
\((3x)(4x) + (3x)(3y) + (4y)(4x) + (4y)(3y)\).
Step 3: Simplify:
\(12x^2 + 9xy + 16xy + 12y^2\).
\(= 12x^2 + 25xy + 12y^2\).
Answer: \(12x^2 + 25xy + 12y^2\)

ii.  \((4a-5b)(3a+2b)\)

Step 1: Expand using distributive property.
\((4a)(3a) + (4a)(2b) + (-5b)(3a) + (-5b)(2b)\).
Step 2: Simplify:
\(12a^2 + 8ab – 15ab – 10b^2\).
\(= 12a^2 – 7ab – 10b^2\).
Answer: \(12a^2 – 7ab – 10b^2\)

iii.  \((2y+z)(7z-3y)\)

Step 1: Expand terms:
\((2y)(7z) + (2y)(-3y) + (z)(7z) + (z)(-3y)\).
Step 2: Simplify:
\(14yz – 6y^2 + 7z^2 – 3yz\).
\(= -6y^2 + 11yz + 7z^2\).
Answer: \(-6y^2 + 11yz + 7z^2\)

iv.  \((2m-4n)(4m-3n)\)

Step 1: Expand terms:
\((2m)(4m) + (2m)(-3n) + (-4n)(4m) + (-4n)(-3n)\).
Step 2: Simplify:
\(8m^2 – 6mn – 16mn + 12n^2\).
\(= 8m^2 – 22mn + 12n^2\).
Answer: \(8m^2 – 22mn + 12n^2\)

Q9: Find the following products:

i.  \((2pq + 0.1mn)(0.2pq + 3mn)\)

Step 1: Expand using distributive law:
\((2pq)(0.2pq) + (2pq)(3mn) + (0.1mn)(0.2pq) + (0.1mn)(3mn)\)
Step 2: Simplify terms:
\((2pq)(0.2pq) = 0.4p^2q^2\)
\((2pq)(3mn) = 6pqmn\)
\((0.1mn)(0.2pq) = 0.02pqmn\)
\((0.1mn)(3mn) = 0.3m^2n^2\)
Step 3: Combine like terms:
\(0.4p^2q^2 + (6 + 0.02)pqmn + 0.3m^2n^2\)
\(= 0.4p^2q^2 + 6.02pqmn + 0.3m^2n^2\)
Answer: \(0.4p^2q^2 + 6.02pqmn + 0.3m^2n^2\)

ii.  \(\left(\frac{4m}{p} – \frac{0.2n}{q}\right)\left(\frac{3m}{p} + \frac{0.5n}{q}\right)\)

Step 1: Expand terms:
\(\frac{4m}{p}\cdot \frac{3m}{p} + \frac{4m}{p}\cdot \frac{0.5n}{q} – \frac{0.2n}{q}\cdot \frac{3m}{p} – \frac{0.2n}{q}\cdot \frac{0.5n}{q}\)
Step 2: Simplify each term:
\(\frac{4m}{p}\cdot \frac{3m}{p} = \frac{12m^2}{p^2}\)
\(\frac{4m}{p}\cdot \frac{0.5n}{q} = \frac{2mn}{pq}\)
\(-\frac{0.2n}{q}\cdot \frac{3m}{p} = -\frac{0.6mn}{pq}\)
\(-\frac{0.2n}{q}\cdot \frac{0.5n}{q} = -\frac{0.1n^2}{q^2}\)
Step 3: Combine terms:
\(\frac{12m^2}{p^2} + \left(\frac{2 – 0.6}{pq}\right)mn – \frac{0.1n^2}{q^2}\)
\(= \frac{12m^2}{p^2} + \frac{1.4mn}{pq} – \frac{0.1n^2}{q^2}\)
Answer: \(\frac{12m^2}{p^2} + \frac{1.4mn}{pq} – \frac{0.1n^2}{q^2}\)

iii.  \(\left(\frac{3}{4}x – 2pq\right)\left(3x – \frac{4}{5}pq\right)\)

Step 1: Expand terms:
\(\frac{3}{4}x \cdot 3x + \frac{3}{4}x \cdot \left(-\frac{4}{5}pq\right) + (-2pq)(3x) + (-2pq)\left(-\frac{4}{5}pq\right)\)
Step 2: Simplify each term:
\(\frac{3}{4}x \cdot 3x = \frac{9}{4}x^2\)
\(\frac{3}{4}x \cdot \left(-\frac{4}{5}pq\right) = -\frac{3}{5}xpq\)
\((-2pq)(3x) = -6xpq\)
\((-2pq)(-\frac{4}{5}pq) = \frac{8}{5}p^2q^2\)
Step 3: Combine like terms:
\(\frac{9}{4}x^2 + \left(-\frac{3}{5} – 6\right)xpq + \frac{8}{5}p^2q^2\)
\(\;= \frac{9}{4}x^2 – \frac{33}{5}xpq + \frac{8}{5}p^2q^2\)
Answer: \(\frac{9}{4}x^2 – \frac{33}{5}xpq + \frac{8}{5}p^2q^2\)

Q10: Find the following products:

i.  \((2a^2+3b^2)(3a^2+2b^2)\)

Step 1: Expand using distributive law:
\((2a^2)(3a^2) + (2a^2)(2b^2) + (3b^2)(3a^2) + (3b^2)(2b^2)\)
Step 2: Simplify each term:
\((2a^2)(3a^2) = 6a^4\)
\((2a^2)(2b^2) = 4a^2b^2\)
\((3b^2)(3a^2) = 9a^2b^2\)
\((3b^2)(2b^2) = 6b^4\)
Step 3: Combine like terms:
\(6a^4 + (4 + 9)a^2b^2 + 6b^4\)
\(= 6a^4 + 13a^2b^2 + 6b^4\)
Answer: \(6a^4 + 13a^2b^2 + 6b^4\)

ii.  \((x^2+5y^2)(y^2-2x^2)\)

Step 1: Expand terms:
\((x^2)(y^2) + (x^2)(-2x^2) + (5y^2)(y^2) + (5y^2)(-2x^2)\)
Step 2: Simplify each term:
\((x^2)(y^2) = x^2y^2\)
\((x^2)(-2x^2) = -2x^4\)
\((5y^2)(y^2) = 5y^4\)
\((5y^2)(-2x^2) = -10x^2y^2\)
Step 3: Combine like terms:
\(-2x^4 + (1 – 10)x^2y^2 + 5y^4\)
\(= -2x^4 – 9x^2y^2 + 5y^4\)
Answer: \(-2x^4 – 9x^2y^2 + 5y^4\)

iii.  \((3c^2-4d^2)(4c^2-3d^2)\)

Step 1: Expand terms:
\((3c^2)(4c^2) + (3c^2)(-3d^2) + (-4d^2)(4c^2) + (-4d^2)(-3d^2)\)
Step 2: Simplify each term:
\((3c^2)(4c^2) = 12c^4\)
\((3c^2)(-3d^2) = -9c^2d^2\)
\((-4d^2)(4c^2) = -16c^2d^2\)
\((-4d^2)(-3d^2) = 12d^4\)
Step 3: Combine like terms:
\(12c^4 + (-9 -16)c^2d^2 + 12d^4\)
\(= 12c^4 – 25c^2d^2 + 12d^4\)
Answer: \(12c^4 – 25c^2d^2 + 12d^4\)