Prove that the following two sets are the same. s1 = {a + bx : a, b ∈ r} = all polynomials which can expressed as a linear combination of 1 and x; s2 = {ax + b(2 + x) : a, b ∈ r} = all polynomials which can expressed as a linear combination of x and 2 + x.

Answer with Step-by-step explanation:

We are given that twos sets

={a+bx:}=All polynomials which can expressed as  a linear combination of 1 and x.

={ax+b(2+x):}=All polynomials which can be expressed as   a linear combination of x and 2+x.

We have to prove that given two sets are same.

={ax+2b+bx}={(a+b)x+2b}={cx+d}

={cx+d}=All polynomials which can be expressed as  a linear combination of 1 and x.

Because a+b=c=Constant

2b= Constant=d

Hence, the two sets are same .

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step-by-step explanation:

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