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# just a few questions

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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 \$a\$ be a constant. What is the largest possible degree of \$f(x) + a•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•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) • g(x)\$?

Find \$t\$ if the expansion of the product of \$x^3 - 4x^2 + 2x - 5\$ and \$x^2 + tx - 7\$ has no \$x^2\$ term.

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?

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? \[(x^4 + x^3 + x^2 + x + 1)^4\]

Feb 17, 2018

#1
+94526
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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)\$?

The x^3 coefficient will have a sum of  1 + 5  = 6

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•g(x)\$?

Depends upon the value of "a"

Note that if   a  = -1/2, the sum of the first two terms in each polynomial is 0....and the sum  of the second two terms in each polynomial is also 0....so the sum will produce a degree of 1.....any other value of "a" will produce a 4th degree polynomial

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•g(x)\$?

Just like the last, if "b"  is -1/2,  the smallest possible degree will be 1

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\$?

Guess that we have a third degree polynomial of the form  ax^3 + bx^2 + cx + d

If  f(0)  =  47, then d =  47

And we have this system

a + b + c + 47  = 32

8a + 4b + 2c + 47  = -13

27a + 9b + 3c + 47  = 16

The solution to this is    a = 52/3, b = -67, c = 104/3, d = 47

And the sum of these is    32

Feb 18, 2018
edited by CPhill  Feb 18, 2018
#2
+94526
+1

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

The degree will  be the result of the product of  x^4 * 2x^4  =  2x^8  ⇒   degree 8

Find \$t\$ if the expansion of the product of \$x^3 - 4x^2 + 2x - 5\$ and \$x^2 + tx - 7\$ has no \$x^2\$ term.

The  x^2 term in the product  will be given by

( 28 + 2t - 5)      so

28 + 2t - 5  = 0

2t  + 23  = 0

2t  = -23    ⇒    t   =  -23/2

Feb 18, 2018
edited by CPhill  Feb 18, 2018
#3
+94526
+1

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 polynomial 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

_____________________________

- 1x^4  + 2x^3 + 8x^2

- 1x^4  -  2x^3  - 3x^2

__________________

4x^3 + 11x^2 + 18x

4x^3 +  8x^2  + 12x

_________________

3x^2  + 6x  + 9

3x^2  + 6x  + 9

_____________

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Feb 18, 2018
#4
+94526
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What is the coefficient of x in (x^4 + x^3 + x^2 + x + 1)^4

Here's the expansion

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  answer to the last question  is   20

Feb 18, 2018