Express f(x) in the form f(x)=(x-k) q(x)+r for the given value of k. f(x)=4 x^4-3 x^3-20 x^2-x ;

step 1 Hello, I hope you're doing well. So we have our f of x function right here, our k value here, an

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

Key Concepts

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear factor (x - k), the remainder of this division is equal to f(k). This theorem provides a quick way to determine the remainder without performing complete division and underpins the rationale behind techniques such as synthetic division.

Division Algorithm for Polynomials

The Division Algorithm for Polynomials states that for any two polynomials f(x) and d(x) (with d(x) not equal to zero), there exist unique polynomials q(x) (the quotient) and r(x) (the remainder) such that f(x) = d(x) * q(x) + r(x), where the degree of r(x) is strictly less than the degree of d(x). In cases where d(x) is a linear polynomial, this framework simplifies the process of polynomial division.

Synthetic Division

Synthetic Division is a streamlined method for dividing a polynomial by a linear divisor of the form (x - k). It allows for a quicker computation of the quotient and remainder compared to traditional long division. This method involves using the value k to systematically process the coefficients of the dividend polynomial, reducing the division process to a series of simple arithmetic operations.

Transcript

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

00:02 So we have our f of x function right here, our k value here, and we need to get our f of x function in the form of x minus k times q of x plus r.

00:13 So to do that, we need to divide this f of x function by x minus k.

00:24 We need to divide this by x minus k.

00:26 And k in this case is 3.

00:28 This is going to be divided by x minus 3.

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

00:34 But the one thing you may notice is that for our numerator, we don't have a constant at the end here.

00:39 We've got x to the fourth, x cubed, x squared, and x, but no constant.

00:43 But we need a constant here at the end in order to do synthetic division.

00:47 So to do that, we're just going to add a plus zero there at the end.

00:51 So now we're going to write down all of the coefficients of our numerator.

00:55 So we've got our coefficients are 4, minus 3, minus 20.

01:03 That's a minus 1.

01:04 And zero.

01:08 And then we're dividing this by our k value, which is three.

01:13 So using synthetic division to divide these, we can bring down this four, four times three is 12, 12 minus or minus three plus 12 is nine, nine times three is 27, minus 20 plus 27 is seven, seven times three is 21, minus one plus 21 is 20, 20 times three is 60, and zero plus 60 is 60.

01:35 So this rate here, here is our remainder term.

01:40 This is our constant term, or x is 0.

01:43 This is our x -term, this is our x -squared term, and this is our x -cubed term.

01:47 So writing out the solution to this synthetic division, we have 4x cubed, plus 9x squared, plus 7x, plus 20, and plus a remainder of 60 over our denominator, which is x -minus 3.

02:10 So that's just plus 6.

02:12 C over x minus 3.

02:14 Remember, this is equal to this synthetic division right here...

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