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 .

the ratio of white tiles to blue tiles is 48: 16. the ratio in it's simplest form is 3: 1.

step-by-step explanation:

ratio: take the first number and add a colon next to it, then add the second number after that. make sure the ratio order matches up with the written problem!

simplest form: we can simplify the ratio 48 : 16 by dividing both terms by the greatest common factor (gcf).

the gcf of 48 and 16 is 16.

divide both terms by 16:

48 ÷ 16 = 3

16 ÷ 16 = 1

reducing to simplest form:

48 : 16 = 3 : 1