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gregori [183]
3 years ago
5

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

Mathematics
1 answer:
igor_vitrenko [27]3 years ago
6 0
<span>tep  1  :</span> 3 Simplify — x <span>Equation at the end of step  1  :</span> 3 px - ((9x + —) + 4) = 0 x <span>Step  2  :</span>Rewriting the whole as an Equivalent Fraction :

<span> 2.1 </span>  Adding a fraction to a whole 

Rewrite the whole as a fraction using <span> x </span> as the denominator :

9x 9x • x 9x = —— = —————— 1 x

<span>Equivalent fraction : </span>The fraction thus generated looks different but has the same value as the whole 

<span>Common denominator : </span>The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

<span> 2.2 </span>      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:

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

<span> 3.1 </span>  Adding a whole to a fraction 

Rewrite the whole as a fraction using <span> x </span> as the denominator :

4 4 • x 4 = — = ————— 1 x <span>Step  4  :</span>Pulling out like terms :

<span> 4.1 </span>    Pull out like factors :

  <span> 9x2 + 3</span>  =  <span> 3 • (3x2 + 1)</span> 

Polynomial Roots Calculator :

<span> 4.2 </span>   Find roots (zeroes) of :      <span> F(x) = 3x2 + 1</span>
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 <span> 1. 

 </span>The factor(s) are: 

of the Leading Coefficient : <span> 1,3 
 </span>of the Trailing Constant : <span> 1 

 </span>Let us test ....

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


Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

<span> 4.3 </span>      Adding up the two equivalent fractions 

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

<span> 5.1 </span>  Subtracting a fraction from a whole 

Rewrite the whole as a fraction using <span> x </span> as the denominator :

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

<span> 5.2 </span>    Factoring <span> 9x2 + 4x + 3</span> 

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

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

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

<span><span>     -27   +   -1   =   -28</span><span>     -9   +   -3   =   -12</span><span>     -3   +   -9   =   -12</span><span>     -1   +   -27   =   -28</span><span>     1   +   27   =   28</span><span>     3   +   9   =   12</span><span>     9   +   3   =   12</span><span>     27   +   1   =   28</span></span>


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

Adding fractions that have a common denominator :

<span> 5.3 </span>      Adding up the two equivalent fractions 

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

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

Now,to get rid of the <span>denominator, </span>Tiger multiplys both sides of the equation by the denominator.

Here's how:

<span> px2-9x2-4x-3 ———————————— • x = 0 • x x </span>

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

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

Solving a Single Variable Equation :

<span> 6.2 </span>    Solve  <span> <span>px2-9x2-4x-3</span> </span> = 0 

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

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