if f(x) ax^2+bx+c , g(x) :1/3x^2+2 and fg(x)= 18x^4+24x^2+11/(3x^2+2)^"
FIND A, B AND C
The expression ends with (3x^2 + 2)^ and lacks an exponent. Maybe ^" is some special notation?
I assume you mean g(x) = 1/(3x^2 + 2)
and fg(x) means f(x)*g(x) and not f(g(x))
As usual, a little ambiguity delays responses.
To find the product of two polynomials \( f(x) \) and \( g(x) \), you need to multiply each term of \( f(x) \) with each term of \( g(x) \) and then combine like terms.
Given that \( f(x) = ax^2 + bx + c \) and \( g(x) = \frac{1}{3}x^2 + 2 \), we can calculate the product \( fg(x) \) by multiplying the terms:
\( fg(x) = (ax^2 + bx + c) \cdot (\frac{1}{3}x^2 + 2) \)
To multiply these two polynomials, distribute each term of the first polynomial with each term of the second polynomial:
\( fg(x) = (\frac{1}{3}ax^4 + 2ax^2) + (bx^2 + 2bx) + (cx^2 + 2c) \)
Now, combine like terms:
\( fg(x) = \frac{1}{3}ax^4 + bx^2 + cx^2 + 2ax^2 + 2bx + 2c \)
Simplify further:
\( fg(x) = \frac{1}{3}ax^4 + (b + 2a)x^2 + 2bx + 2c \)
So, the product of \( f(x) \) and \( g(x) \) is:
\( fg(x) = \frac{1}{3}ax^4 + (b + 2a)x^2 + 2bx + 2c \)