Question

let $f(x)=2x^2 - 8x + 5$ and $g(x)=x^2 + 4$. find the following.
a) $(f + g)(x)$
b) $(f - g)(x)$
c) $(f \cdot g)(x)$

Explanation:

Step1: Add the two functions

$(f+g)(x) = (2x^2 - 8x + 5) + (x^2 + 4)$
Combine like terms:
$(2x^2 + x^2) - 8x + (5 + 4) = 3x^2 - 8x + 9$

Step2: Subtract the functions

$(f-g)(x) = (2x^2 - 8x + 5) - (x^2 + 4)$
Distribute the negative sign and combine like terms:
$2x^2 - 8x + 5 - x^2 - 4 = (2x^2 - x^2) - 8x + (5 - 4) = x^2 - 8x + 1$

Step3: Multiply the two functions

$(f \cdot g)(x) = (2x^2 - 8x + 5)(x^2 + 4)$
Use distributive property (FOIL extended):
$2x^2(x^2 + 4) - 8x(x^2 + 4) + 5(x^2 + 4)$
$= 2x^4 + 8x^2 - 8x^3 - 32x + 5x^2 + 20$
Combine like terms:
$2x^4 - 8x^3 + (8x^2 + 5x^2) - 32x + 20 = 2x^4 - 8x^3 + 13x^2 - 32x + 20$

Answer:

a) $3x^2 - 8x + 9$
b) $x^2 - 8x + 1$
c) $2x^4 - 8x^3 + 13x^2 - 32x + 20$