Fundamental Operations on Algebraic Expressions (Addition, Subtraction, Production, and Division) of Algebraic Expressions and Remainder Theorem

Fundamental Operations on Algebraic Expressions (Addition, Subtraction, Production, and Division) of Algebraic Expressions

Addition of Algebraic Expressions

• By joining like words with the aid of their coefficients and applying the associative property, commutative property, and/or distributive property of multiplication, we can create an algebraic expression.
• The horizontal or column methods are also acceptable (vertical method).

Example:
$(4x^2$ $+3y^2$ $-6x$ $+4y$ $-2)$,  $(2x$ $-y^2$ $+3x^3$ $-4y$ $+3)$,  and  $(-6x^2$ $-2y^2$ $-1)$

Solution:

2^nd Method

Horizontal method

$(4x^2$ $+3y^2$ $-6x$ $+4y$ $-2)$ $+(-3x^2$ $-y^2$ $+2x$ $-4y$ $+3)$ $+(-6x^2$ $-2y^2$ $-1)$

$(4-3-6)x^2$ $+(3-1-2)y^2$ $+(-6+2)x+(4-4)$ $+(-2+3-1)$

$(4-9)x^2$ $+(3-3)y^2$ $+(-4)x$ $+(0)y$ $+(3-3)$

$-5x^2$ $-4x$  Ans.

 

Subtraction of Algebraic Expression

• When combining two algebraic expressions, the sign of each term in the subtracted expression must be changed before adding the result to the original equation.

E.g.

$1)$ If $A$ $=x^2$ $+y^2$ $-z^2$, $B$ $=3x^2$ $-2y^2$ $+5z^2$ and $C$ $=3y^2$ $-5x^2$ $-z^2$

    Find the value of $2A$ $-3B$ $+4B$

Solution:

$2A-3B+4C$ $=2(x^2+y^2-z^2)$ $-3(3x^2-2y^2+5z^2)$ $+4(3y^2-5x^2-z^2)$

$=2x^2$ $+2y^2$ $-2z^2$ $-9x^2$ $+6y^2$ $-15z^2$ $+12y^2$ $-20x^2$ $-4z^2$

$2A$ $-3B$ $+4C$ $=-27x^2$ $+20y^2$ $-21z^2$    Ans.

 

Product of Algebraic Expression

• The laws of exponents, rules of signs, and the commutative, associative, and distributive characteristics of multiplication over addition are used to execute multiplication.

Find the product
$\left( 2x^{1/2}y^{3/8}+5xy^{2/9} \right)$ $\left( 6x^{2}y^{3}-3x^{3/2}y^{7/2} \right)$

      Solution:

$\left( 2x^{1/2}y^{3/8}+5xy^{2/9} \right)$ $\left( 6x^{2}y^{3}-3x^{3/2}y^{7/2} \right)$

$2x^{1/2}y^{3/8}$ $\left( 6x^{2}y^{3}-3x^{3/2}y^{7/2} \right)$ $+5xy^{2/9}$ $\left( 6x^2y^3-3x^{3/2}y^{7/2} \right)$

$12x^{1/2+2}y^{3/8+3}$ $-6x^{1/2+3/2}y^{3/8+7/2}$ $+30x^{1+2}y^{2/9+3}$ $-15x^{1+3/2}y^{2/9+7/2}$

$5x^{5/2}y^{27/8}$ $-6x^{2}y^{31/8}$ $+30x^{3}y^{29/9}$ $-15x^{5/2}y^{67/10}$   Ans.

 

Division of Algebraic Expression

Division in arithmetic

$9÷2$ $=4+1$ $\to$ with remainder $1$

Each segment of a division is given a name.

$9÷2$ $=4\to$ Quotient with remainder $1$

$9$ $\to$ Dividend  and  $2$ $\to$ Divisor

Quotient

• The answer to a division problem

Dividend

• A number is divided by another number.

 

Divisor

• A number to just be divided into another number

Remainder

• The amount or number left over after the division
• We can also divide polynomials
• If we divide a polynomial with a polynomial the quotient also be a polynomial

$f(x)$ $÷g(x)$ $=q(x)$ with remainder $r(x)$

 

Problem:

1)   Divide $(x^3$ $-19x$ $-30)$ by $(x^2$ $-3x$ $-10)$

Solution:

 

image

• A constant will be left behind when a polynomial is divided by a linear polynomial or a polynomial of degree $1$.
  
  
  

The Remainder Theorem

If a polynomial $p(x)$ of degree $n$, $(n\ge 1)$ is divided by a linear polynomial $(x$ $-a)$, then the remainder $r$ $=p(a)$

Proof:

If we divide a polynomial $p(x)$ by another polynomial $d(x)$ to find a quotient polynomial $q(x)$ and a remainder polynomial $r(x)$ such that:

 a)    $P(x)$ $=d(x).q(x)+r(x)$
 b)    The degree of $r(x)$ is less than that of $d(x)$

Since our divisor $(x-r)$ is of degree $1$, the remainder must be of degree zero, that is a constant,

Hence write

$P(x)$ $=(x-r)q(x)+r$  $(1)$

where $r$ is a constant

Since $(1)$ is an identity and therefore, true for all values of $x$, it is true in particular
When $x$ $=a$
By substituting $x$ $=a$ in $(1)$

$P(a)$ $=(a-a)q(x)+r$

$P(a)$ $=0.q(x)+r$

$P(a)$ $=r$    Which is what we lead to prove

Thus, if the divisor is linear, the above theorem provides an efficient way of finding the remainder without being involved in the process of long division.

Remarks:

1. If $r$ $=0$ then $(x-a)$ is a factor of $P(x)$
2. If $(x-a)$ is a factor of $P(x)$, then $a$ is a root of $P(x)$ $=0$
3. If $P(x)$ is a polynomial and $a$ is a real number with $P(x)$ $=0$, then $a$ is a root of the polynomial equation $P(x)$ $=0$.

Problem:1
Find the remainder by means of the remainder theorem $x^4$ $-2x^2$ $+3x$ $+3$ is divided by $x$$ -3$
solution:

Let $P(x)$ $=x^4$ $-2x^2$ $+3x$ $+3$
and $x-3$ $=0$

$x$ $=3$  $x-3$ comparing with $x-a$

$x-3$ $=x-a$
$-3$ $=-a$
$a$ $=3$

$P(3)$ $=(3)^4$ $-2(3)^2$ $+3(3)$ $+3$
$P(3)$ $=81$ $-2(9)$ $+9$ $+3$
$P(x)=$ $81$ $-18$ $+12$

$P(x)$ $=75$

$r$ $=3$  Ans.

Problem 2.

Decide whether or not the statement is true or not $x+3$ is a factor of $x^3-x^2-22x+24$

Solution:
Let $P(x)=3x^3-x^2-22x+24$        $(1)$
If $P(a)=0$ then $a$ is a root of $P(x)$
$x+3=0$ then $x=-3$ put in $(1)$

$P(-3)=3(-3)^3-(-3)^2-22(-3)+24$

$P(-3)=3(-27)-(9)+66+24$

$P(-3)=-81-9+90$

$P(-3)=-90+90=0$

$P(-3)=0$

Hence $x+3$ is a factor of $x^3-x^2-22x+24$

$r=0$        Ans.