Multiplication and Division of Algebraic expressions

In multiplication of algebraic Expressions, we use the laws of exponents.

**Monomial –** an algebraic expression consisting of one term e.g. $5x$ is a monomial consisting of two factors $5$ and $x$

**Binomial **– an algebraic expression consisting of two terms e.g. $5x + 3y$ is a binomial expression consisting of two terms $5x$ and $3y$ each with two factors.

**Trinomial –** an algebraic expression consisting of three terms e.g. $5x^3 + 3y – 2z$

**Polynomial –** an algebraic expression consisting of more than two terms i.e. an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).

There are different types of multiplication that will be covered in this section.

**Monomial by monomial**

Multiply $y^3$ and $y^{-4}$

Solution:

$= y^3 * y^{-4}$

$= y^{3+(-4)}$

$= y^{-1}$

Find the product of $p^2$ and $p^5$

Solution:

$= p^2$ * $p^5$

$= p^{2+5}$

$= p^7$

Find the product of $3ab$ and $4a^{2}b$

Solution :

$= 3ab * 4a^{2}b$

$= (3 * 4) * a * a^2 b * b$

$= 12a^{1 +2}b^{1 +1}$

$= 12a^3b^2 $

**Monomial by Binomial**

Multiply: $4abc$ and $5a^{2}b -3c^6$

Solution :

$= 4abc * 5a^{2}b -3c^6$

$= ( 4 * 5)(abc * a^{2}b -3c^6 )$

$= 20 a^{1+ 2} * b^1 + (-3) * c^{1 + 6} $

$= 20a^3b^{-2}c^7 $

$= \frac{20a^3c^7}{b^2}$ [ Make the exponent of $b$ positive]

**Binomial by Binomial**

Multiply:(2x -1)(5x +6)

Solution :

$= (2x * 5x)+ (2x * 6) + ({-1} * 5x)+ ({-1} * 6)$

$= 10x^2 + 12x – 5x -6$ (add like terms)

$= 10x^2 + 7x – 6$

Multiply: $(8x – 3)( 9x -2)$

Solution:

$= (8x * 9x)+ (8x * -2)+ (-3 * 9x) + (-3 * -2)$

$= 72x^2 – 16x – 27x + 6$

$=72x^2 – 43x + 6$

$(y + 2)( y – 2)$

Solution:

$= (y * y)+ (7 * -2)+ (2 * y)+ (2 * -2)$

$= y^2 + 2y – 2y – 4 $

$= y^2 + 0 – 4$

$= y^2 – 4$

$(x^2 + 5)(x^2 + 10)$

Solution:

$= (x^2 * x^2 )+ (x^2 * 10)+ (5 * x^2 )+ (5 * 10)$

$= x 2 + 2 + 10×2 + 5×2 + 50 $

$= x 4 + 15x 2 + 50 $

$(2x^2 + 8)(x^2 – 7)$

Solution:

$= (2x^2 * x^2 )+ (2x^2 * (-7)+ (8 * x^2 )+ (8 * (-7)$

$= 2x^2 + 2 – 14x^2 + 8x^2 -56 $

$= 2x 4 – 6x 2 – 56$

**Polynomial by Binomial**

Multiply $(a^2 + ab + b^2)(a – b) $

Solution:

$(a^2 + ab + b^2 )(a – b)$

Use a distributive property.

$= a^2 (a – b) + ab( a – b) + b^2 (a – b) $

$= a^2 * a – a^2 * b + ab * a – ab * b + b^2 * a – b^2 * b$

$= a^3 – a^{2}b + a^2b – ab^2 + ab^2 – b^3$[ Add like terms]

$= a^3 -b^3 $

$(1 – 4x)( 1 + x + x^2)$

Solution:

$= 1( 1 + x + x^2 ) – 4x ( 1 + x + x^2 )$

$= 1 + 1*x + 1*x^2 – 4x*1 – 4x*x – 4x*x^2$

$= 1 + x + x 2 – 4x – 4x 2 – 4x 3$ [ Bring the like terms together]

$= 1 + x – 4x + x^2 – 4x^2 – 4x^3$

$= 1 – 3x -3x^2 – 4x^3$[ Add like terms]

$= {-4}x^3 – 3x^2 -3x + 1$ [ arranging in descending order of exponents]

Multiply: $(a^2 – b^2) ( 4a^3 – b^3)$

Solution:

$=(a^2 – b^2 )( 4a^3 – b^3 )$

$= a^2 (4a^3 – b^3 )- b^2 (4a^3 – b^3 )$

$= 4a^5 – a^2 b^3 – 4a^3b^2 + b^5$

There are different types of division that will be covered in this section.

**Monomial by Monomial **

$\frac{5x^4}{x^2} $

$= 5x^{4-2} $

$= 5x^2$

$\frac{5x^5}{10x^2}$

$ = \frac{1}{2} x^{5-2}$

$= \frac{1}{2} x^3$

**Binomial by Monomial **

\item $\frac{x+3}{x} $

$= \frac{x}{x}$ + $\frac{3}{x}$

$ = 1 + \frac{3}{x}$

$\frac{x^2+3}{x} $

$= \frac{x^2}{x}$+ $\frac{3}{x}$

$ = x + \frac{3}{x}$

$ \frac{3x^4}{x^2 + x} $

$= (\frac{3x^4}{x+1})(\frac{1}{x})$

$=\frac{3x^3}{x+1}$

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