A cubic polynomial $x^{3} + a x^{2} + b x + c$ has roots 1, 2, and 3. Find the coefficients $a$ , $b$ , and $c$ .

hard 3 marks Class 9 · Mathematics · Polynomials

Answer

a = - 6,; b = 11,; c = - 6

The correct answer is $a = - 6,; b = 11,; c = - 6$ — Thus $a = - 6$ , $b = 11$ , $c = - 6$ .

Step-by-step solution

By the factor theorem, the polynomial equals $(x - 1) (x - 2) (x - 3)$ .
First multiply $(x - 1) (x - 2) = x^{2} - 3 x + 2$ .
Now multiply by $(x - 3)$ : $(x^{2} - 3 x + 2) (x - 3)$ .
Expand: $x^{3} - 3 x^{2} - 3 x^{2} + 9 x + 2 x - 6 = x^{3} - 6 x^{2} + 11 x - 6$ .
Match with $x^{3} + a x^{2} + b x + c$ .
Thus $a = - 6$ , $b = 11$ , $c = - 6$ .

Common mistake: Incorrect sign for

a

when expanding

Concept tested: Coefficients from roots