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1.   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)?

2.   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 $$f(x) + a\cdot g(x)?$$

3.  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 $$f(x) + b\cdot g(x)$$?

4.  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?

Nov 18, 2019

#1
+1

also

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

Nov 18, 2019
#2
+19719
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1>  Well, yo would just add the x^3 tems in the two fxns   1 + 5 = 6

2>  If a is a constant, it will only change the coefficients of the functions, not the degree of the terms

so the greatest degree will still be 4    form   the x^4  and 2x^4 in the two fxns

3>  Similar to 2   except the lowest degree will be  1    from the 2x   term

4>    not sure

the constant will be 47

5>   when the two functions are multiplied you will have a x^4 * 2x^4    = 2 x^8 term in the final answer....    dgree 8

Nov 18, 2019
#3
+106515
+1

1.   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 coefficient  is  just   the  sum of 1x^3 + 5x^3  = 6x^3  =  6

2.   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)

As long as  "a"  is not  -1/2 , the  sum will result in a polynomial of degree 4

3.  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) ?

If "b"  is  -1/2     the resulting sum will be a first power  polynomial....this is the smallest possible degree

4.  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?

If   f(0) = 47.....then the constant term of the polynomial must just be  47

If  f(1)  =  32

Then the sum of the coefficients  is just   32 - 47  =   - 15

Nov 18, 2019
edited by CPhill  Nov 18, 2019
edited by CPhill  Nov 18, 2019