In Exercises 41–43, divide using long division. (10 x^3-26 x^2+17 x-13) ÷(5 x-3)

Name: Problem 42, Chapter 3: Precalculus
Uploaded: 2020-03-05T18:25:50-08:00
Duration: 3 min 35 s
Channel: Thomas Emment
Description: In Exercises 41–43, divide using long division. (10 x^3-26 x^2+17 x-13) ÷(5 x-3)

In Exercises 41–43, divide using long division. (10 x^3-26...

Key Concepts

Handling Leading Coefficients

In polynomial division, dividing the leading term of the dividend by the leading term of the divisor yields the next term in the quotient. This term is then used to multiply the entire divisor, and its product is subtracted from the dividend, systematically eliminating the highest-degree term and simplifying the division process.

Polynomial Long Division

Polynomial long division is a technique used to divide one polynomial by another, mirroring the long division process for numbers. It involves dividing the highest degree terms, determining quotient terms, multiplying back, and subtracting to progressively reduce the degree of the remaining polynomial.

Division Algorithm for Polynomials

This algorithm states that given a dividend polynomial and a non-zero divisor polynomial, there exist unique quotient and remainder polynomials such that the dividend equals the product of the divisor and the quotient plus the remainder. The degree of the remainder is always less than the degree of the divisor, ensuring a proper division.

Transcript

00:01 Okie dokey.

00:02 We want to perform long division on this equation here, 10x cube minus 26x square, plus 17x minus 13, and we want to divide it by 5x minus 3.

00:13 But we're doing so via long division.

00:16 And i'm just going to zoom out here because when they say long division, they mean long, okay? how we're going to go about this is we're going to just see how much of our first term can go into each time.

00:30 Turn by turn making them equal to zero okay so how many five x is can we get into 10x cubed well simply two x squared so we'll put our two x squared up here and we'll take away what we had so five x it's going to be minus 10 x cubed and now here's the interesting bit minus three times two x squared that's going to be minus six x squared we can't remember we do have a second term here and what we do with one we'll have to do with the other okay so 10x cubed minus 10x cube very simply zero that's what we were hoping for should know that and then we've got minus 26x squared minus minus 6x squared it's going to be plus so we'll end up with minus 20 x squared i hope you can see that minus 26 x squared plus 6x squared minus 20x squared plus 17 x minus 13 always do remember to bring your terms down.

01:33 Okay, so next up we've got minus 20x.

01:35 How can we make minus 20x, sorry, minus 20x squared, how can we make minus 20x squared with 5x? well, 4xs.

01:43 So we've got plus 4x here.

01:47 And if you take the term away, we've got, oh, sorry, it's minus 20x squared.

01:52 So to make minus 20x squared out of our 5xes, we'll have to do minus 4x.

01:58 My bad there, sorry about that so we'll end up with minus minus 20x squared and minus 3 times minus 4x well simply plus 12x okay so minus 20x squared plus 20x squared 0 17x minus 12 x that's 5x so drop your term down 5x 13...

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