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# polynomials

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(a) Let b be a constant. What is the smallest possible degree of the polynomial $$f(x)+b\cdot g(x)$$, where $$f(x) = 2x^5 - 6x^4 - 4x^3 + 12x^2 + 7x - 5$$and $$g(x) = x^5 - 3x^4 - 2x^3 - 6x^2 + 14x - 10$$?

(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 $$(x^3 + x^2 + x + 1)^7$$?

(d) There exists a polynomial f(x) and a constant k such that
$$(x^2 - 2x - 5) f(x) = 2x^4 - x^3 + kx^2 - 19x - 10.$$What is k?

Mar 12, 2024

### 1+0 Answers

#1
+129690
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(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

Mar 12, 2024