Question

18. write a quadratic polynomial such that it is divided by X-1,x-2x-3 leaves remainder 1, 2 and 4
respectively is.​

Answer 1

Step-by-step explanation:

know that the remainder when f(x) is divided by x−α is f(α). If we say that

f(x)=ax2+bx+c

then we have

a×12+b×1+ca+b+ca×22+b×2+c4a+2b+ca×32+b×3+c9a+3b+c=11=11=22=22=37=37(1)(2)(3)

We then have 3 equations in 3 unknowns so we can solve these for a,b,c: (2)−(1):

3a+b=11(4)

(3)−(2):

5a+b=15(5)

(5)−(4):

2a=4a=2

From (4):

3×2+b=11b=5

From (1):

2+5+c=11c=4

And therefore, our quadratic is:

f(x)=2x2+5x+4

– lioness99a

Answer 2

Step-by-step explanation:

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

Let our quadratic expression be [Math Processing Error]

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Now, [Math Processing Error]

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

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Now, [Math Processing Error]

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We can thus rewrite the expression as:

[Math Processing Error]

[Math Processing Error]

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

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Also, [Math Processing Error]

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We can thus rewrite the expression as:

[Math Processing Error]

[Math Processing Error]

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.

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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: [Math Processing Error]