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Let f(x) = x^3 + 3x ^2 + 4x - 7 and g(x) = -7x^4 + 5x^3 +x^2 - 7. What is the coefficient of x^3 in the sum f(x) + g(x)?

Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b\cdot g(x)$?

 

Suppose $f$ is a polynomial such that $f(0) = 47$, $f(1) = 32$, $f(2) = -13$, and $f(3)=16$. What is the sum of the coefficients of $f$?

Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. What is the degree of $f(x) \cdot g(x)$?

There is a polynomial which, when multiplied by $x^2 + 2x + 3$, gives $2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9$. What is that polynomial? (Give your answer as a polynomial in which the terms appear in order of decreasing degree. In other words, "$x^5 + x + 2$" is a valid answer, but "$2+x^5+x$" is not.)

What is the coefficient of $x$ in $(x^4 + x^3 + x^2 + x + 1)^4$?

What is the coefficient of $x^3$ in this expression?\[(x^4 + x^3 + x^2 + x + 1)^4\]

 

 

 


any help would be appreciated!

 Jul 22, 2020
 #1
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(3) 

Suppose f(x) is a cubic polynomial y = ax^3 + bx^2 + cx + d.  Then

d = 47

a + b + c + d = 32

8a + 4b + 2c + d = -13

27a + 9b + 3c + d = 16

==> f(x) = 46/3*x^3 - 57x^2 + 68/3*x + 47.

 

The sum of the coefficients is 46/3 - 57 + 68/3 = -19.

 

(4) Matching coefficients, the polynomial is 2x^3 + x^2 - 2x + 3.

 Jul 22, 2020
 #2
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it says they are wrong :/ I'm still trying to figure it out

Guest Jul 24, 2020

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