Express f(x) in the form f(x)=(x-k) q(x)+r for the given value of k. f(x)=3 x^4+4 x^3-10 x^2+15

step 1 Hello, hope you're doing well. This is our f of x function right here, and this is our k value.

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

Key Concepts

Polynomial Division

Polynomial division is the process of dividing a polynomial by another, yielding a quotient and a remainder. In the division of a polynomial f(x) by a divisor of the form (x - k), the result can be written as f(x) = (x - k)q(x) + r, where q(x) is the quotient polynomial and r is the remainder, which is a constant when the divisor is linear.

Synthetic Division

Synthetic division is a simplified method used to divide a polynomial by a linear factor of the form (x - k). It efficiently produces the coefficients of the quotient polynomial and the remainder, reducing the computational complexity compared to long division, especially when dealing with higher degree polynomials.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear factor (x - k), the remainder of that division is equal to the value of the polynomial evaluated at k, i.e., r = f(k). This theorem provides a quick way to determine the remainder without performing the complete division process.

Transcript

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

00:02 This is our f of x function right here, and this is our k value.

00:05 We need to get this f of x function in the form x minus k times q of x plus r.

00:15 So did you say we were going to take this f of x function right here, and we're going to divide it by x minus k.

00:27 Okay in this case is equal to minus 1, so x minus k will be x minus minus 1, so that'll just be x plus 1.

00:34 So we're going to divide these two using a synthetic division.

00:40 So first, before we start that, you may notice that for this numerator, we're missing a term with a term for x here in the numerator.

00:51 We've got x to the fourth, x cubed, x squared, and a constant, but we're missing our x term.

00:55 We need an x term, though, in order to perform synthetic division.

01:01 So we're just going to add in a zero x right here.

01:04 So now we're going to write out all the coefficients of our numerator.

01:08 So we've got 3, 4, minus 10, got a 0 here for our x coefficients, and we've got 15 here.

01:20 We are going to divide this by our k value of minus 1.

01:25 So starting out with some, to start out, we're going to bring down this 3 here.

01:33 Then we've got 3 times minus 1, gives us minus 3.

01:36 4 minus 3 is 1.

01:37 1 times minus 1 gives us minus 1, minus 10 minus 1 gives us minus 11, minus 11 times minus 1 gives us 11, 0 plus 11 is 11, 11 times minus 1 gives us minus 11, and 15 minus 11 is 4.

01:53 So now, so this 4, this value right here is our remainder value.

01:59 This is just our constant here, our x of the 0 value.

02:03 This is our coefficient for x.

02:05 This is our x squared coefficient...

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