1. Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $a$ be a constant. What is the largest possible degree of $f(x) + a\cdot g(x)$?
2. 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?
3. What is the coefficient of $x$ in $(x^4 + x^3 + x^2 + x + 1)^4$?
4. What is the coefficient of $x^3$ in this expression? \[(x^4 + x^3 + x^2 + x + 1)^4\]
+128474 1. \(f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1 \)
Let a be a constant
What is the highest pssible degree of
\(f(x) + a\cdot g(x) \)
Multiplying g(x) by a non-zero constant will not change the degree of g(x)
As long as "a" is not equal to -1/2, the degree of f(x) + a*g(x) will be 4
+128474 2. 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?
We can find this with division
2x^3 - x^2 + 4x + 3
x^2 + 2x + 3 [ 2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9 ]
2x^5 + 4x^4 + 6x^3
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-x^4 + 2x^3 + 8x^2
-x^4 - 2x^3 - 3x^2
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4x^3 + 11x^2 +18x
4x^3 + 8x^2 + 12x + 9
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3x^2 + 6x + 9
3x^2 +6x + 9
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The polynomial we need is in red
+128474 3. What is the coefficient of x in (x^4 + x^3 + x^2 + x + 1)^4?
x^16 + 4 x^15 + 10 x^14 + 20 x^13 + 35 x^12 + 52 x^11 + 68 x^10 + 80 x^9 + 85 x^8 + 80 x^7 + 68 x^6 + 52 x^5 + 35 x^4 + 20 x^3 + 10 x^2 + 4 x + 1
The coefficient is 4
4. From above....the coefficient on the x^3 term is 20
