Question 1149382
Find the quadratic which has a remainder of -6 when divided by x - 1, a
remainder of -4 when divided by x -3 and no remainder when divided by x + 1.
<pre>
Let the quadratic polynomial be Ax² + Bx + C


The remainder theorem says this:

If you substitute r into a polynomial, you will get the same answer as you will
get if you divide the polynomial by x-r and take the remainder.

>>Find the quadratic which has a remainder of -6 when divided by x - 1<<

So if r=1, then the remainder theorem says this:

If you substitute 1 into a polynomial, you will get the same answer as you will
get if you divide the polynomial by x-1 and take the remainder.

So let's substitute 1 into the quadratic polynomial and set it equal to the
remainder -6

A(1)² + B(1) + C = -6
       A + B + C = -6

>>a remainder of -4 when divided by x - 3<<

So if r=3, then the remainder theorem says this:

If you substitute 3 into a polynomial, you will get the same answer as you will
get if you divide the polynomial by x-3 and take the remainder.

So let's substitute 3 into the quadratic polynomial and set it equal to the
remainder -4

A(3)² + B(3) + C = -4
     9A + 3B + C = -4

>>and no remainder when divided by x + 1<<

Note that x + 1 is the same as x - (-1) 

So if r=-1, then the remainder theorem says this:

If you substitute -1 into a polynomial, you will get the same answer as you will
get if you divide the polynomial by x+1 and take the remainder.

So let's substitute -1 into the quadratic polynomial and set it equal to the
remainder 0.

A(-1)² + B(-1) + C = 0
         A - B + C = 9
 
So we have 3 equations in 3 unknowns:

 A +  B + C = -6
9A + 3B + C = -4
 A -  B + C =  0

Solve that and get A=1, B=-3, and C=-4

So Ax² + Bx + C becomes

    x² - 3x - 4  

Checking:

     <u>        x - 2</u>
x - 1) x² - 3x - 4
       <u>x² -  x</u>
           -2x - 4
           <u>-2x + 2</u>
                -6   <--the remainder is -6 when we divide by x - 1.

 
     <u>        x - 0</u>
x - 3) x² - 3x - 4
       <u>x² - 3x</u>
            0x - 4
            <u>0x + 0</u>
                -4   <--the remainder is -4 when we divide by x - 3.

     <u>        x - 4</u>
x + 1) x² - 3x - 4
       <u>x² +  x</u>
           -4x - 4
           <u>-4x - 4</u>
                 0   <--the remainder is 0 (no remainder) when we divide by x - 1.

Edwin</pre>