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多項式的問題，高手請幫忙
發問:
4.(a)若 (Ax + 1)(x – A)(x + B) + A(x + 2) + 1=2x^ + 3x^2 – 7x – 1，試不展開恆等式的左方，求A 和B 的值。 (b)由此，判斷下列各多項式是否2x^3 + 3x^2 – 9x – 6 的因式。 (i)x^2 + x – 6 (ii)2x^2 + 7x + 3 (iii)2x^2 – x – 1
最佳解答:
a. (Ax + 1)(x - A)(x + B) + A(x + 2) + 1 = 2x3 + 3x2 - 7x - 1 L.H.S. = (Ax + 1)(x - A)(x + B) + A(x + 2) + 1 = (Ax2 - A2x + x - A)(x + B) + Ax + 2A + 1 = Ax3 + ABx2 - A2x2 - A2Bx + x2 + Bx - Ax - AB + Ax + 2A + 1 = Ax3 + (AB - A2 + 1)x2 + (-A2B + B)x + (2A - AB + 1) Comparing the coefficient of the term x3, A = 2 Comparing the coefficient of the term x2, (2)B - (2)2 + 1 = 3 B = 3 b. From (a), L.H.S. = R.H.S. ∴ (2x + 1)(x - 2)(x + 3) + 2(x +2) + 1 = 2x3 + 3x2 - 7x - 1 (2x + 1)(x - 2)(x + 3) = 2x3 + 3x2 - 9x - 6 Choice (i): x2 + x - 6 = (x - 2)(x + 3) ∴ x2 + x - 6 is a factor of 2x3 + 3x2 - 9x - 6. Choice (ii): 2x2 + 7x + 3 = (2x + 1)(x + 3) ∴ 2x2 + 7x +3 is a factor of 2x3 + 3x2 - 9x - 6. Choice (iii): 2x2 - x - 1 = (2x + 1)(x - 1) ∴ 2x2 - x - 1 is not a factor of 2x3 + 3x2 - 9x - 6.
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