Algebra 2 reviews

Introduction:

Polynomials are algebraic expressions with no division by a variable.

For ex:The expression is 3x2-2x+5 is an example of polynomial.

Consider the algebraic expression `(7x+3)/(y)` which is not a polynomial as the variable ‘y’ appears in the denominator or there is a division by a variable.


Wecan perform all mathematical operations like addition, subtraction, multiplication, division, exponentiation etc on polynomials. When one polynomial is divided by another polynomial we get like numeric division, we get a quotient and a remainder. Remainder theorem helps us to calculate the remainder easily in some special cases without actuallyperforming the Division of Polynomials.


Remainder Theorem


Remaindertheorem says when an polynomial f(x) is divided by (x-a), the remainderis f(a). Let us illustrate this with an example

Ex 1 : Consider function f(x) = 3x2+4 is divided by x-3, the remainder is f(3)

Sol: Step 1:  Plug in x = 3 in  f(x) = 3x2+4

 Step 2:  f(3)= 3x32+4 = 31

Nowlet us consider a situation where f(x) is divided by (x+a) then the remainder can be found using the same formula by putting (x+a) as (x- (-a)).

So,dividing f(x) by (x+a) is equivalent to dividing f(x) by (x- (-a)). So as per theorem the remainder is f(-a). Let us solve an example here:


Ex 2: Find the remainder when f(x) = 7x2+4x+6 is divided by (x+3)

Sol: Step 1:  f(x) = 7x2+4x+6  is divided by (x+3) or (x- -3). So, the remainder is f(-3)

 Step 2:  Plug in x = (-3) in f(x) = 7x2+4x+6

  f(-3) = 7x(-3)2+4x(-3)+6=63-12+6=57

NowConsider the case f(x) is divided by (ax+b). In this we need to make itan fashion equal to f(x) divided by (x-a). This can be done as

`(f(x))/(ax+b) = (f(x))/(a(x-(-b/a)))` . So the remainder is f ( `(-b)/(a)` )

Ex3: Find the remainder when f(x) = 3x2-4x+3 is divided by (3x+2)

Sol: Step 1: The given division is (3x2-4x+3) is divided by (3x+2).

  This is equivalent to dividing`(3x^(2)-4x+3)/(3 )` by `(x -(-2/3))`. So, the remainder is f(`(-2)/(3)` )

 Step 2: f(`(-2)/(3)`) = 3`(-2/3)^2`–4(`(-2)/(3)`)+3 = `(4)/(3)`  -`(-8/3)`+3 = 4+3 =7


Exercises


Prob 1: Find the remainder when 7x3-4x2+8x+9 is divided by 7x-5

Ans: Step 1: As seen above, when f(x) is divided by (ax+b), the remainder is f(-b/a).

  Step 2: When 7x3-4x2+8x+9 is divided by 7x-5, the remainder is f(`(5)/(7)`)

 Step 3: f((`(5)/(7)`)= 7((`(5)/(7)`)3-4((`(5)/(7)`)2+8((`(5)/(7)`)+9 = (`(746)/(49))` 


Prob 2: For what value of a will f(x) = 5x2-6x+a be completely divisible by (x-5)?

Ans: Step 1:  If f(x) is completely divisible by (x-5) then the remainder is zero. I.e. f(5)=0

  Step 2:  f(5)= 5(5)2-6(5)+a=0 

  125-30+a=0

 Step 3:  95+a=0

 a=-95

5x2-6x-95 is completely divisible by 95


Prob 3: : For what value of a will f(x) = 5x2-6x+a when divided  by (x-5) will leave a remainder of 6?

Ans: Step 1: If there is a reminder of 6 then f(x)-6 will be completely divisible by (x-5)

  So f(5)-6=0

 Step 2:  5x52-6x5+a-6=0 

  125-30+a-6=0

 Step 3: 89+a=0

  a="-89

When 5x2-6x-89 is divided by (x-5), the remainder is 6