Divide using synthetic division. (x^2+3) ÷(x-1)

Name: Problem 19, Chapter 6: Algebra 2
Uploaded: 2020-01-19T21:27:42-08:00
Duration: 3 min 25 s
Channel: Karuna Ramroop
Description: Divide using synthetic division. (x^2+3) ÷(x-1)

Key Concepts

Handling Missing Terms

When performing polynomial division, especially using synthetic division, it is crucial to account for all degrees of the variable by inserting zero coefficients for any missing terms. This ensures that the placement of coefficients aligns correctly with the corresponding powers of the variable, which is essential for accurate computation.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder of the division is equal to f(c). This concept not only provides a quick check on the division process but also helps in understanding the relationship between polynomial division and evaluating polynomials at specific points.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the process by focusing on the coefficients of the polynomial rather than writing out all the variable terms, making calculations quicker and reducing the possibility of arithmetic errors.

Polynomial Division

Polynomial division is analogous to the long division of numbers, where one polynomial (the dividend) is divided by another polynomial (the divisor) to yield a quotient and possibly a remainder. It is a fundamental process used to simplify expressions, find factors, and solve polynomial equations.

Transcript

00:01 All right, so we need to use synthetic division to figure out what x squared plus 3 divided by x minus 1 will give us.

00:13 So what we're going to do is we've done synthetic division a few times before.

00:19 So what we're going to do is we're first going to take our 0, x equals 1, our hopeful 0.

00:27 As long as we don't get a remainder, we'll know that's what it is.

00:29 We're going to take that and we're going to put it up in our little corner.

00:34 But now here's something new that we haven't seen before.

00:37 We have x squared plus 3.

00:40 There are some missing terms in between them, one missing term specifically.

00:46 Since this is a degree 2 polynomial, i should also see a degree 1 and degree 0 polynomial.

00:53 Degree 0 just means that it's just a number, like 3.

00:58 So i have my degree zero term, but where is my degree one term? well, my degree one term is just 0x.

01:09 Right, we have 1x squared plus 0x plus 3.

01:14 Normally we don't write this because there's no need to.

01:18 But when we're dividing, this is very important because with long division, we need placeholders and with synthetic division, we need to know what our coefficients are.

01:27 Because all we're doing with synthetic division is we're taking this degree two polynomial, dividing it by a degree one polynomial, and we know we're going to end up with a degree one polynomial after.

01:41 But we don't know what the coefficients of that polynomial are.

01:45 So we are going to take the coefficients of our polynomial.

01:52 So we know that we have an invisible one in front of the x square.

01:57 We know that the coefficient in front of x is just zero, and that's why it wasn't written there...

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