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Multiplication and Division of algebraic expressions

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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Edzai Conilias Zvobwo is passionate about empowering Africans through mathematics, problem-solving techniques and media. As such, he founded MathsGee. Through this organisation, he has helped create an ecosystem for disseminating information, training, and supporting STEM education to all African people. A maths evangelist who teaches mathematical thinking as a life skill, Edzai’s quest has seen him being named the SABC Ambassador for STEM; he has been invited to address Fortune 500 C-suite executives at the Mobile 360 North America; was nominated to represent Southern Africa at the inaugural United Nations Youth Skills Day in New York; was invited to be a contributor to the World Bank Group Youth Summit in 2016; has won the 2014 SADC Protocol on Gender and Development award for his contribution to women’s empowerment in education; and has partnered with local and global firms in STEM interventions.