As a Rule 1: To multiply a monomial to a
multinomial, simply multiply
each term of the
multinomial by the monomial. This is a direct application of distributive property.
Rule 2: To multiply a multinomial to another multinomial we apply the distributive property by multiplying the whole multiplicand by each term of the multiplier.
Hence, we introduce the symbol xn where n is the exponent and x is the base. Noted that xn is the result of multiplying x n times, that is,
x . x = x2 x
multiplied twice
x . x . x = x3 x multiplied thrice
x . x ….xn = xn x multiplied n times
In multiplying x to itself, its exponent are added. But, if x is multiplied to a variable other than x, exponent not added. The result is the product of the two.
1.) xn . xm = xn+m
2.) xn . ym = xnym
To illustrate the above statement, we have;
Example: x4 . x7 = x4+7 = x11 - x multiply itself, exponent is added.
Example: the product of -3x3 and 6x4 is
(-3x3) (6x4) = -18x3+4 = -18x7
Example: The product of 3x -3y5z and 20x2y3z-4
(3x -3y5z) (20x2y3z -4) = 60x -3+2y5+3z1+(-4) = 60x-1y8z-3
-- x multiplied to another variable, exponent not added
Apply Rule 1:
Example: The product of 2x and 2x2 – 2x + 5 is
2x(2x2 – 2x + 5) = 4x3 – 2x2 + 10x
Apply Rule 2:
Example: Get the product of 3x + 2 and 2x2 – 5x + 1
By distributive property ,
(3x + 2) (2x2 – 5x + 1) = 6x3 - 15x2 + 3x + 4x2 – 10x + 2
= 6x3 – 11x2 – 7x + 2 ans.
-15x2 + 4x2 = -11x2
3x - 10x = -7x
Other form, 3x(2x2 – 5x + 1) + 2(2x2 – 5x + 1)
= 6x3 - 15x2 + 3x + 4x2 – 10x + 2
= 6x3 – 11x2 – 7x + 2 ans.
Check: 2x2 – 5x + 1
3x + 2
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6x3 - 15x2 + 3x
4x2 – 10x + 2
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6x3 – 11x2 – 7x + 2 ans.