Divide using synthetic division. (x^3-7 x^2-7 x+20) ÷(x+4) |...

Name: Problem 15, Chapter 6: Algebra 2
Uploaded: 2020-01-19T21:17:46-08:00
Duration: 3 min 34 s
Channel: Karuna Ramroop
Description: Divide using synthetic division. (x^3-7 x^2-7 x+20) ÷(x+4)

Key Concepts

Remainder Theorem

The Remainder Theorem states that when a polynomial is divided by a linear divisor of the form x - c, the remainder is equal to the value of the polynomial evaluated at x = c. This theorem not only provides a quick way to determine the remainder without performing full division but also builds a connection between polynomial division and function evaluation.

Polynomial Division

Polynomial division is the process of dividing a polynomial by another polynomial, similar to numerical long division. It involves breaking down the dividend into simpler parts that can be divided by the divisor, resulting in a quotient and possibly a remainder. This concept is fundamental in algebra for simplifying expressions, solving equations, and understanding the behavior of polynomial functions.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form x - c. Instead of writing out a full long division, synthetic division uses only the coefficients of the polynomial, applying a series of multiplications and additions to efficiently determine the quotient and remainder. This method simplifies computations and is particularly useful for higher degree polynomials.

Transcript

00:01 All right, so we've been asked to do this division using synthetic division.

00:08 So what we're going to do with synthetic division, we know that what we have to do is pretend that this x plus 4 is actually a zero or root of our function.

00:22 We have to pretend that x equals negative 4 is a zero of my polynomial.

00:30 So the idea is that if this division works out to have no remainder, then if i were to have factored this polynomial x plus 4 would show up in there somewhere.

00:43 So we're going to see whether that's true or not.

00:46 So we're first going to put our negative 4 up in this corner right here, and then we're going to take the coefficients on each of our terms, and we're going to bring those straight here.

00:59 So that's an invisible 1.

01:01 There's a negative 7, another negative 7, and a 20.

01:07 So what we're going to do is we're first going to bring our 1 straight down.

01:13 We don't do anything with that.

01:17 And this is a little bit hard to remember.

01:19 What you can always think is once i see a number down here that i just wrote here, my next step is always to multiply it by my number up in the corner.

01:28 So i'm going to take this one, and i'm going to multiply it by negative 4.

01:31 And that is going to give me negative 4, which i will bring over to the right on my next column.

01:38 So now, every time we have two numbers lined up like this, all we do is add them...

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