Under normal conditions, a reply would not be so complete as is the one that follows. However, Purplemath does not, to my knowledge, currently offer tutorials on these particular topics. Therefore, I will provide a more extensive response.
cruxxfay wrote:"Find the polynomial f such that f(2x-1) = 16x^4 - 32x^3 +12x^2"
The result polynomial is the composition of the unknown f(x) with g(x) = 2x - 1. By nature of composition of functions
, we know that (f o g o g-1
)(x) = f(x). Since g(x) = 2x - 1, then g-1
(x) = (x + 1)/2. We are given that (f o g)(x) = 16x4
. Combining these results, we find:. . . . .. . . . . . .
Simplify to obtain the required polynomial answer.
cruxxfay wrote:"When polynomial f is divided by (x-3) the remainder is 2, and when divided by (x+1) the remainder is -4. Find the remainder when polynomial f is divided by (x^2 -2x - 3)"
The Factor Theorem
states that any polynomial f(x) may be restated as p(x)q(x) + r(x), where p(x) is the (polynomial) divisor, q(x) is the result (that is, the quotient), and r(x) is the remainder. The Remainder Theorem
, related to the Factor Theorem, states that the value of a polynomial f(x) at x = a is the same as the value of the remainder when f(x) is divided by x - a. We use these as follows:
We are given that f(x) must be divisible by the given quadratic. Therefore:. . . . .
Immediately, we note that the specified linear divisors are factors of the quadratic divisor. Continuing directly and applying the Remainder Theorem, we find:. . . . .
So r(3) = 2.. . . . .
So r(-1) = -4. Since we are dividing by a quadratic, then logically the remainder can have a degree of 0 or 1. Since r(x) is clearly not constant, then r(x) must be linear. Then r(x) = ax + b for some values a and b. One may then find the equation of the remainder as being r(x) = (3/2)x - (5/2).
Since f(x) = (x2
- 2x - 2)p(x) + r(x), then the remainder, upon division by (x2
- 2x - 2), must be r(x).
Kindly please reply with queries, should any of the above leave questions in your mind. Thank you.