To find the two polynomials whose difference is [tex]\(2x - 5\)[/tex], let's denote them by [tex]\(P(x)\)[/tex] and [tex]\(Q(x)\)[/tex]. We know that:
[tex]\[ P(x) - Q(x) = 2x - 5 \][/tex]
We will express both polynomials in general form as linear polynomials, since [tex]\(2x - 5\)[/tex] is a linear polynomial:
[tex]\[ P(x) = a_1 x + b_1 \][/tex]
[tex]\[ Q(x) = a_2 x + b_2 \][/tex]
We are given that their difference is:
[tex]\[ (a_1 x + b_1) - (a_2 x + b_2) = 2x - 5 \][/tex]
By distributing the subtraction, we get:
[tex]\[ a_1 x + b_1 - a_2 x - b_2 = 2x - 5 \][/tex]
To solve for the coefficients, we can equate the corresponding terms on both sides of the equation. Equate the coefficients of [tex]\(x\)[/tex]:
[tex]\[ a_1 - a_2 = 2 \][/tex]
And equate the constant terms:
[tex]\[ b_1 - b_2 = -5 \][/tex]
Now, we need to find values for [tex]\(a_1\)[/tex], [tex]\(a_2\)[/tex], [tex]\(b_1\)[/tex], and [tex]\(b_2\)[/tex] that satisfy these equations. One way to approach this is to choose specific values for [tex]\(a_1\)[/tex] and [tex]\(b_1\)[/tex], and then solve for [tex]\(a_2\)[/tex] and [tex]\(b_2\)[/tex].
Let's start by choosing [tex]\(a_1 = 2\)[/tex] and [tex]\(b_1 = -5\)[/tex]:
1. Using [tex]\(a_1 = 2\)[/tex], substitute into the first equation:
[tex]\[ 2 - a_2 = 2 \][/tex]
Solve for [tex]\(a_2\)[/tex]:
[tex]\[ a_2 = 0 \][/tex]
2. Using [tex]\(b_1 = -5\)[/tex], substitute into the second equation:
[tex]\[ -5 - b_2 = -5 \][/tex]
Solve for [tex]\(b_2\)[/tex]:
[tex]\[ b_2 = 0 \][/tex]
Thus, one possible pair of polynomials that satisfies the conditions is:
[tex]\[ P(x) = 2x - 5 \][/tex]
[tex]\[ Q(x) = 0 \][/tex]
Hence, the two polynomials are [tex]\(P(x) = 2x - 5\)[/tex] and [tex]\(Q(x) = 0\)[/tex].
