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| 1. |
x*4–3x*2+4x+5 is dovided by x*2+1–x |
| Answer» We have, f(x) =x4\xa0- 3x2 + 4x + 5 and g(x) = x2\xa0+ 1 - x.We find that degree f(x) = 4 and degree g(x) = 2.Therefore, quotient q(x) is of degree 4 - 2 = 2 and remainder r(x) is of degree less than 2 = degree (g(x)).So, let q(x) = ax2 +bx + c and r(x) = px + q.Using division algorithm, we have f(x) = g(x) {tex}\\times{/tex}\xa0q(x) + r(x){tex}\\Rightarrow{/tex}x4 + 0x3 - 3x2 + 4x + 5 = (x2 - x + 1){tex}\\times{/tex}(ax2 + bx + c) + (px + q){tex}\\Rightarrow{/tex}x4 + 0x3 - 3x2 + 4x + 5 = ax4 - ax3 + ax2 + bx3 - bx2 + bx + cx2 - cx + c + px + q{tex}\\Rightarrow{/tex}\xa0x4 + 0x3 - 3x2 + 4x + 5 = ax4\xa0+ (b - a)x3 + (c - b + a)x2 + (b - c + p)x + c + qOn equating the coefficients of various powers of x on both sides, we geta = 1 [On equating the coefficients of x2]b - a = 0 [On equating the coefficients of x3]c - b + a = -3 [On equating the coefficients of x2]b - c + p = 4 [On equating the coefficient of x]and, c + q = 5 [On equating the constant terms]a = 1....... (i)a = 1 put in b- a = 0b - 1 = 0b = 1....... (ii)c - b + a = -3c - 1 + 1 = -3c = -3.......... (iii)b - c + p = 41 - (-3) + p = 41 + 3 + p = 4p = 0 ......... (iv)c + q = 5- 3 + q = 5q = 5 + 3 = 8 ....... (v)From (i), (ii) , (iii), (iv) and (v), we get a = 1, b = 1, c = -3, p = 0 and q = 8Therefore, Quotient q(x) = x2 + x - 3 and Remainder r(x) = 8 | |