Understanding Polynomial Factors

The video tutorial explains how to determine if x-3 is a factor of a fourth-degree polynomial using algebraic long division and the polynomial remainder theorem. It demonstrates that if the remainder is zero when the polynomial is evaluated at x=3, then x-3 is a factor. The tutorial includes a detailed computational example to illustrate the process.

What is the main method discussed to determine if x-3 is a factor of the polynomial?

• Factoring by grouping

• Synthetic division

• Graphical analysis

• Algebraic long division

According to the polynomial remainder theorem, what should be the remainder when a polynomial is divided by x-a?

• The polynomial evaluated at x=0

• The derivative of the polynomial

• Zero

• The polynomial evaluated at x=a

In the context of the polynomial remainder theorem, what is the value of 'a' when checking if x-3 is a factor?

• 3

• -3

• 0

• 1

What is the first step in evaluating the polynomial at x=3?

• Subtract 3 from each term

• Multiply the coefficients by 3

• Calculate 3 to the fourth power

• Add all coefficients

What is the result of 3 to the fourth power?

• 243

• 9

• 81

• 27

What is the product of 11 and 27 in the calculation?

• 135

• 243

• 270

• 297

What is the sum of 162 and 135 in the arithmetic calculation?

• 135

• 297

• 270

• 243

What does the cancellation of the last two terms in the calculation indicate?

• Addition of a new factor

• Change in polynomial degree

• Error in calculation

• Simplification of the expression

What is the final remainder when the polynomial is evaluated at x=3?

• One

• Negative one

• Zero

• Three

What conclusion can be drawn if the remainder is zero?

• x-3 is not a factor

• x-3 is a factor

• The polynomial has no real roots

• The polynomial is prime