Express f(x) in the form f(x)=(x-k) q(x)+r for the given value of k. f(x)=-x^3+x^2+3 x-2 ; k=2

Name: Problem 22, Chapter 3: Precalculus Student Solutions Manual 5th
Uploaded: 2020-08-05T13:41:47-08:00
Duration: 3 min 22 s
Channel: Cullen Miller
Description: Express f(x) in the form f(x)=(x-k) q(x)+r for the given value of k. f(x)=-x^3+x^2+3 x-2 ; k=2

step 1 Hello, hope you're doing well. So we have our F of X function here. We have our K value here. An

Express f(x) in the form f(x)=(x-k) q(x)+r for the given...

Key Concepts

Division Algorithm for Polynomials

This principle states that for any two polynomials, where the divisor is nonzero, it is possible to express the dividend as the product of the divisor and a quotient polynomial plus a remainder polynomial. The degree of the remainder is less than the degree of the divisor. This is analogous to the division algorithm for integers.

Quotient and Remainder

In polynomial division, the quotient is the polynomial obtained when the dividend is divided by the divisor, and the remainder is what is left over. The relationship is represented as f(x) = (divisor) * (quotient) + (remainder), which provides a unique decomposition of the polynomial.

Synthetic Division

Synthetic division is a streamlined alternative to long division when dividing a polynomial by a linear factor of the form (x - k). It simplifies the process by focusing only on the coefficients of the polynomial, making it faster and more efficient, especially for finding the quotient and remainder.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - k), the remainder is equal to f(k). This theorem provides a quick method to evaluate the remainder without performing full polynomial division, linking polynomial evaluation to division.

Transcript

00:01 Hello, hope you're doing well.

00:02 So we have our f of x function here.

00:04 We have our k value here.

00:06 And we need to get our f of x function in the form x minus k, once q of x plus r.

00:17 So in order to do this, we're going to divide this f of x function.

00:25 We're going to divide it by x minus k.

00:27 So k is 2 in this case.

00:29 It's going to be divide this by x minus 2.

00:31 So we're going to use synthetic division to divide this.

00:34 So we're going to write out all of the coefficients.

00:36 Of the numerator.

00:38 So we've got, this is a minus one and this is a one.

00:40 So all our coefficients are going to be minus one, one, three, and minus two.

00:49 Then we're going to divide this by our k value, which is just two.

00:56 So using synthetic division to divide these, we are going to carry this minus one down.

01:02 We have minus one times two is minus two.

01:05 One minus two is minus one.

01:06 Minus 1 times 2 is minus 2, 3 minus 2 is 1, 1 times 2 is 2, and minus 2 plus 2 is 0.

01:14 So this is our remainder value, this is our constant value, our x to the 0 value, this is our x value, and this is our x squared value.

01:23 So writing out the solution to this synthetic division, we're left with minus 1x squared or minus x squared, minus 1x, or just minus x, then plus our constant 1.

01:34 Then we have no remainder here, our remainder is zero, so we don't have to worry about that.

01:40 Remember from before, this is equal to this division right here...

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