Q no 2 find a and b

Q no 2 find a and b 2. bx2- 7x+9) 38k' —22x2+37x +45.

We have (x^{3} + {bx}^{2} - 7 x + 9) (x^{2} + ax + 5) = x^{5} + 13 x^{4} + 38 x^{3} - 22 x^{2} + 37 x + 45 Consider LHS (x^{3} + {bx}^{2} - 7 x + 9) (x^{2} + ax + 5) = x^{3} (x^{2} + ax + 5) + {bx}^{2} (x^{2} + ax + 5) - 7 x (x^{2} + ax + 5) + 9 (x^{2} + ax + 5) = x^{5} + {ax}^{4} + 5 x^{3} + {bx}^{4} + {abx}^{3} + 5 {bx}^{2} - 7 x^{3} - 7 {ax}^{2} - 35 x + 9 x^{2} + 9 ax + 45 = x^{5} + (a + b) x^{4} + (5 + ab - 7) x^{3} + (5 b - 7 a + 9) x^{2} + (- 35 + 9 a) x + 45 = x^{5} + (a + b) x^{4} + (ab - 2) x^{3} + (5 b - 7 a + 9) x^{2} + (- 35 + 9 a) x + 45 On comparing with RHS, we get x^{5} + (a + b) x^{4} + (ab - 2) x^{3} + (5 b - 7 a + 9) x^{2} + (- 35 + 9 a) x + 45 = x^{5} + 13 x^{4} + 38 x^{3} - 22 x^{2} + 37 x + 45 a + b = 13 . . . (1) - 35 + 9 a = 37 . . . (2) Using (2), we get - 35 + 9 a = 37 \Rightarrow 9 a = 37 + 35 \Rightarrow 9 a = 72 \Rightarrow a = 8 From (1) \Rightarrow a + b = 13 \Rightarrow b = 13 - a \Rightarrow b = 13 - 8 \Rightarrow b = 5