You will learn:
- Special products
- Product of two Binomials
- Product of Sum and Difference of two terms
- Expansions
The multiplication of certain types of algebraic expressions can be obtained mentally by using some rules. Such multiplications are termed as special products.
Products of two Binomials
Let us find products of \((x+y)(x+z)\),
Here \((x + y)(x + z) = x(x + z)+y(x + z) = x^2 + xz + yx + yz\)
\(x^2 +x(y + z) + yz\)
Hence, \((x + y)(x + z) = x^2 +(y+ z)x + yz\)
In the same way, we can easily obtain the following special products:
\((x + y)(x - z) = x^2 +(y - z)x - yz\)
\((x - y)(x + z) = x^2 +(z - y)x - yz\)
\((x - y)(x - z) = x^2 - (y+z)x + yz\)
Examples: Find the following products-
- \((2a+3)(2a+4) = (2a)^2 + (3+4)(2a) + (3)(4) = 4a^2 + 14a + 12\)
- \((2m - p^2)(2m + q^2) = (2m)^2 + (q^2 - p^2)(2m) - (q^2)(p^2) = 4m^2 + 2m(q^2 - p^2) - q^2p^2\)
Product of Sum and Difference of two terms
\((x + y)(x - z) = x^2 + (y - z)x -yz\)
replace z by y
\(⇒ (x + y)(x - y) = x^2 - y^2 \)
Expansions
When an algebraic expression is multiplied by itself to its second, third, or any other power then the process is termed expansion.
\((x +y)^2 = x^2 + 2xy + y^2 \)
\((x - y)^2 = x^2 - 2xy + y^2\)
\((x + y+ z)^2 = x^2 + y^2 + z^2 + 2(xy+yz+xz)\)
\((x + y)^3 = x^3+y^3+ 3xy(x+y) \)
\((x - y)^3 = x^3 - y^3 - 3xy(x - y)\)
Examples:
\((\sqrt2 + x)^2 = (\sqrt2)^2 + 2 \cdot \sqrt2.x + x^2 \\ 2+2\sqrt2x+x^2\)
\((104)^2 = (100+4)^2 = (100)^2 + 2⋅ 4 ⋅100+ 42 = 10000 + 800 + 16 =10816 \)
Perfect Square Trinomial
Any trinomial that can be expressed as \( (x^2 + 2xy + y^2) \textrm{ or } (x^2 - 2xy + y^2)\) is known as a perfect square trinomial.
\(x^2 + 2xy + y^2\) is a perfect square of (x+y) and \(x^2 - 2xy + y^2\) is perfect square of (x−y).
