Find f(x) and g(x) so the function can be expressed as y = f(g(x)). y =2/x^(2) + 3
2 answers:
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Answer:
See Below.
Step-by-step explanation:
By the Factor Theorem, if we divide <em>q(x)</em> into <em>p(x) </em>and the resulting remainder is 0, then <em>p(x)</em> is divisible by <em>q(x)</em> (i.e. there are no remainders).
Problem 1)
We are given:
We should find the remainder when dividing <em>p(x)</em> and <em>q(x)</em>. We can use the Polynomial Remainder Theorem. When dividing a polynomial <em>p(x)</em> by a binomial in the form of (<em>x</em> - <em>a</em>), then the remainder will be <em>p(a).</em>
Here, our divisor is (<em>x</em> + 1) or (<em>x</em> - (-1)). So, <em>a </em>= -1.
Then by the Polynomial Remainder Theorem, the remainder when performing <em>p(x)/q(x)</em> is:
The remainder is 0, satisfying the Factor Theorem. <em>p(x)</em> is indeed divisible by <em>q(x)</em>.
Problem 2)
We are given:
Again, use the PRT. In this case, <em>a</em> = 3. So:
It satisfies the Factor Theorem.
Problem 3)
We are given:
Use the PRT. In this case, <em>a</em> = 10. So:
It satisfies the Factor Theorem.
Since all three cases satisfy the Factor Theorem, <em>p(x)</em> is divisible by <em>q(x)</em> in all three instances.