Question

Find the quadratic polynomial when divided by
X, X - 1 and x 2 leaves remainders 1, 2 and 9
respectively.​

Answer 1

Step-by-step explanation:

What is the quadratic polynomial when divided by x,x-1,x-2 leaves remainders 1,2 and 9, respectively?

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I’m sure this isn’t the most elegant solution, but I think it’s quite easy to follow.

Let our quadratic expression be ax2+bx+c

Now, ax2+bx+c=(ax+b)x+c

The first part is clearly divisible by x, leaving the remainder c. Thus c = 1.

Now, x=(x−1)+1⇒x2=[(x−1)+1]2

=(x−1)2+2(x−1)+1

We can thus rewrite the expression as:

a(x−1)2+2a(x−1)+a+b(x−1)+b+1

=a(x−1)2+(2a+b)(x−1)+(a+b+1)

The first two parts are clearly divisible by (x-1), leaving the remainder a + b + 1 = 2. This implies that a + b = 1.

Also, x=(x−2)+2⇒x2=[(x−2)+2]2

=(x−2)2+4(x−2)+4

We can thus rewrite the expression as:

a(x−2)2+4a(x−2)+4a+b(x−2)+2b+1

=a(x−2)2+(4a+b)(x−2)+(4a+2b+1)

The first two parts are clearly divisible by (x-2), leaving the remainder 4a + 2b + 1 = 9. This implies that 4a + 2b = 8, thus 2a + b = 4.

We now have a pair of simultaneous equations:

a + b = 1

2a + b = 4

Subtracting Equation 1 from Equation 2, we have a = 3, which means that b = -2.

Answer: 3x2−2x+1