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?

2: Consider the polynomials [f(x)=4x^3+3x^2+2x+1]and [g(x)=3-4x+5x^2-6x^3.] Find c such that the polynomial f(x)+cg(x) has degree 2.

3: What is the coefficient of x^3 when 7x^4-3x^3 -3x^2-8x + 1 is multiplied by 8x^4+2x^3 - 7x^2 + 3x + 4 and the like terms are combined?

Thanks my dudes

AnonymousConfusedGuy
Nov 28, 2017

#1**+1 **

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?

Let us suppose that we have this form

P(x) = ax^3 + bx^2 + cx + d where d = 47

So we have that

a + b + c + 47 = 32 ⇒ a + b + c = -15

8a + 4b + 2c + 47 = -13 ⇒ 8a + 4b + 2c = -60

27a + 9b + 3c + 47 = 16 ⇒ 27a + 9b + 3c = -31

Multiply the first equation by -2 and add it to the second equation

Multiply the first equation by -3 and add it to the third equation

6a + 2b = -30 ⇒ 3a + b = - 15

24a + 6b = 14 ⇒ 12a + 3b = 7

Multiply the first equation by -3 and add it to the second

3a = 52 ⇒ a = 52/3

3(52/3) + b = -15 ⇒ 52 + b = -15 ⇒ b = -67

(52/3) + (-67) + c = -15 ⇒ c = -15 - 52/3 + 67 ⇒ c = 104/3

So.... the sum of the coefficients is

52/3 - 67 + 104/3 = -15

CPhill
Nov 28, 2017

#2**+1 **

Thanks a bunch, but... My math program says that's wrong. I couldn't find anything wrong with your calculation though?

AnonymousConfusedGuy
Nov 28, 2017

#5**0 **

Here is the graph : https://www.desmos.com/calculator/if4wjn6elr

All the specified points appear.....is it possible that the polynomial you want is of a different degree ???

CPhill
Nov 28, 2017

#6**+1 **

ah, I think it's looking for the coefficients of f, not the coefficients of 47, 32, -13, and 16, is it possible that would render a different answer?

AnonymousConfusedGuy
Nov 28, 2017

#3**+1 **

2: Consider the polynomials [f(x)=4x^3+3x^2+2x+1]and [g(x)=3-4x+5x^2-6x^3.] Find c such that the polynomial f(x)+cg(x) has degree 2.

[4x^3+3x^2+2x+1] + c [ 3-4x+5x^2-6x^3]

[4x^3+3x^2+2x+1] + c [ -6x^3 + 5x^2 - 4x + 3 ]

For a resulting polynomial to have degree 2, we must have that

4 - 6c = 0

4 = 6c

4/6 = c = 2/3

So we have the resulting polynomial

[4x^3+3x^2+2x+1] + (2/3) [ -6x^3 + 5x^2 - 4x + 3 ] =

(1/3) (19 x^2 - 2 x + 9)

CPhill
Nov 28, 2017

#4**+1 **

3: What is the coefficient of x^3 when 7x^4-3x^3 -3x^2-8x + 1 is multiplied by 8x^4+2x^3 - 7x^2 + 3x + 4 and the like terms are combined?

[ 7x^4 - 3x^3 -3x^2 - 8x + 1 ] * [ 8x^4 + 2x^3 - 7x^2 + 3x + 4 ]

The coefficient of the x^3 term will come from

(-3 * 4) + (-3 * 3) + (-8 * -7) + (1 * 2) =

-12 - 9 + 56 + 2 =

37

CPhill
Nov 28, 2017