Question

If p(x) = x 2 + 7x + 3 is divided by x + 4, the remainder is

Answer 1

tep  1  : 3 Simplify — x Equation at the end of step  1  : 3 px - ((9x + —) + 4) = 0 x Step  2  :Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  x  as the denominator :

9x 9x • x 9x = —— = —————— 1 x

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole 

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

9x • x + 3 9x2 + 3 —————————— = ——————— x x Equation at the end of step  2  : (9x2 + 3) px - (————————— + 4) = 0 x Step  3  :Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

4 4 • x 4 = — = ————— 1 x Step  4  :Pulling out like terms :

 4.1     Pull out like factors :

   9x2 + 3  =   3 • (3x2 + 1) 

Polynomial Roots Calculator :

 4.2    Find roots (zeroes) of :       F(x) = 3x2 + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  3  and the Trailing Constant is  1. 

 The factor(s) are: 

of the Leading Coefficient :  1,3 
 of the Trailing Constant :  1 

 Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      4.00        -1     3      -0.33      1.33        1     1      1.00      4.00        1     3      0.33      1.33   

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions 

3 • (3x2+1) + 4 • x 9x2 + 4x + 3 ——————————————————— = ———————————— x x Equation at the end of step  4  : (9x2 + 4x + 3) px - —————————————— = 0 x Step  5  :Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a fraction from a whole 

Rewrite the whole as a fraction using  x  as the denominator :

px px • x px = —— = —————— 1 x Trying to factor by splitting the middle term

 5.2     Factoring  9x2 + 4x + 3 

The first term is,  9x2  its coefficient is  9 .
The middle term is,  +4x  its coefficient is  4 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   9 • 3 = 27 

Step-2 : Find two factors of  27  whose sum equals the coefficient of the middle term, which is   4 .

     -27   +   -1   =   -28     -9   +   -3   =   -12     -3   +   -9   =   -12     -1   +   -27   =   -28     1   +   27   =   28     3   +   9   =   12     9   +   3   =   12     27   +   1   =   28

Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions 

px • x - ((9x2+4x+3)) px2 - 9x2 - 4x - 3 ————————————————————— = —————————————————— x x Equation at the end of step  5  : px2 - 9x2 - 4x - 3 —————————————————— = 0 x Step  6  :When a fraction equals zero : 6.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

px2-9x2-4x-3 ———————————— • x = 0 • x x

Now, on the left hand side, the  x  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   px2-9x2-4x-3  = 0

Solving a Single Variable Equation :

 6.2     Solve   px2-9x2-4x-3  = 0 

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.