For the following exercises, use synthetic division to determine whether the first expression is

step 1 Hi guys, welcome back to today's video. Today we're going to take a dive into factorization and

For the following exercises, use synthetic division to...

Key Concepts

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors, which may include linear or higher-degree polynomials. This process is foundational for solving polynomial equations and simplifying expressions by breaking them into simpler multiplicative components.

Synthetic Division

Synthetic division is a streamlined process of dividing a polynomial by a linear divisor of the form x - c. It simplifies the long division process by operating directly on the coefficients, allowing one to quickly determine the quotient and remainder without writing out the full polynomial division.

Remainder Theorem

The remainder theorem states that when a polynomial f(x) is divided by a linear divisor of the form x - c, the remainder is equal to f(c). This concept is useful for quickly checking if a given expression is a factor of a polynomial, as a zero remainder indicates factorization.

Factor Theorem

The factor theorem is a specific case of the remainder theorem which asserts that a polynomial f(x) has a factor (x - c) if and only if f(c) equals zero. This theorem provides a systematic way to determine the factors of a polynomial by evaluating it at specific values.

Transcript

00:01 Hi guys, welcome back to today's video.

00:02 Today we're going to take a dive into factorization and how we can determine if a polynomial is factorable into another polynomial using synthetic division.

00:12 So let's take a dive in.

00:14 So the first thing that we have to keep in mind always is whenever we're doing our divisions, in order to determine if something is factorable, we have to make sure that no remainder should be left.

00:26 So that means we should get zero as our last.

00:30 Value of this entire problem.

00:32 Okay, so let's dive in.

00:34 So today's problem is negative 4x cube plus 5x squared plus 8, all divided by x plus 3.

00:43 So the first thing that i can notice is we're actually missing the coefficient for our x.

00:50 Okay, and whenever that happens, i always love to pretend that we have it there, except our coefficient for x is going to be zero x.

01:00 And i'm just going to keep that for safekeeping and go in order so that it's much easier to convert everything.

01:07 So let's dive in.

01:09 The first thing we want to do is we want to bring our coefficients to our box right here.

01:13 So the coefficient for x cubed is negative 4.

01:18 The coefficient for x squared is 5.

01:21 The coefficient for x is 0.

01:23 And we have our last constant.

01:25 All right.

01:26 So let's bring this negative 4 all the way down.

01:31 And the next thing we have to do is we need to bring this 3 all the way over here.

01:36 And the only way we're allowed to do that is if we make this plus 3 and we multiply it by a negative.

01:43 So when i mean multiply it by a negative, i'm going to take this plus 3, multiply it by a negative.

01:49 We're going to get minus 3...

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