+280 (a) Let b be a constant. What is the smallest possible degree of the polynomial f(x)+b⋅g(x), where f(x)=2x5−6x4−4x3+12x2+7x−5and g(x)=x5−3x4−2x3−6x2+14x−10?
(b) Let f be a cubic polynomial such that f(0) = 5, f(1) = -5, f(2) = 8, and f(3)=13. What is the sum of the coefficients of f?
(c) What is the coefficient of x in (x3+x2+x+1)7?
(d) There exists a polynomial f(x) and a constant k such that
(x2−2x−5)f(x)=2x4−x3+kx2−19x−10.What is k?
+130466 (a)
If b = -2 the addition of the polynomials = 24x^2 -21x + 15 = the smallest possible polynomial
(d)
2x^2 + 3x + (16 + k)
x^2 - 2x - 5 [ 2x^4 - x^3 + kx^2 -19x -10 ]
2x^4 -4x^3 -10x^2
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3x^3 + (10 + k)x^2 -19x
3x^3 -6 x^2 -15x
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(16+ k)x^2 - 4x - 10
(16 + k)x^2 - (32 - 2k)x - 80 - 5k
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(28 + 2k)x + (70 + 5k)
(28 + 2k) + (70 + 5k) = 0
98 = -7k
-98/7 = -14 = k
