Question

Give two trinomials f(x) and g(x), so that their sum equals 3x-2

Answer 1

To find two trinomials \( f(x) \) and \( g(x) \) such that their sum equals \( 3x - 2 \), we can start by expressing \( f(x) \) and \( g(x) \) in the general trinomial form:

\[
f(x) = ax^2 + bx + c
\]
\[
g(x) = dx^2 + ex + f
\]

We want \( f(x) + g(x) = 3x - 2 \). Thus, we can combine these:

\[
f(x) + g(x) = (a + d)x^2 + (b + e)x + (c + f)
\]

This gives us the following system of equations by comparing coefficients with \( 3x - 2 \):

1. \( a + d = 0 \) (coefficient of \( x^2 \))
2. \( b + e = 3 \) (coefficient of \( x \))
3. \( c + f = -2 \) (constant term)

Now, we can choose some values to satisfy these equations.

Let’s choose:
- \( a = 1 \) and \( d = -1 \) (so that \( a + d = 0 \))
- \( b = 2 \) and \( e = 1 \) (so that \( b + e = 3 \))
- \( c = 1 \) and \( f = -3 \) (so that \( c + f = -2 \))

Now we can write the trinomials:

\[
f(x) = 1x^2 + 2x + 1 = x^2 + 2x + 1
\]

\[
g(x) = -1x^2 + 1x - 3 = -x^2 + x - 3
\]

Now, let’s check their sum:

\[
f(x) + g(x) = (x^2 + 2x + 1) + (-x^2 + x - 3)
\]

Combining like terms:

\[
= (x^2 - x^2) + (2x + x) + (1 - 3)
\]
\[
= 0 + 3x - 2
\]
\[
= 3x - 2
\]

Thus, the two trinomials \( f(x) = x^2 + 2x + 1 \) and \( g(x) = -x^2 + x - 3 \) satisfy the requirement that their sum equals \( 3x - 2 \).