[Video] Fundamentals of Factoring

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We go through common prerequisite topics about fundamentals of factoring. We look at common factors and factoring quadratic polynomials.

The following videos will cover topics about factoring polynomials:

Fundamentals of Factoring

Factoring Video Warning!

Introduction and Question 1: Factoring With Common Factors

(Click the arrow to the right to see the Introduction video and first question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Which of the following ways does $2x^2 + 6x$ factor?

$2x(x+3)$ $x+3$ $(2x+1)(x+3)$ $12x^3$

(Click the arrow to the right to see an example.)

Example 1

Checking The Answer

Question 2: Factoring Quadratic Polynomials 1

(Click the arrow to the right to see the second question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Which of the following ways does $x^2-7x+12$ factor?

$(x-6)(x-2)$ $(x+6)(x+2)$ $(x+3)(x+4)$ $(x+3)(x-4)$ $(x-3)(x+4)$ $(x-3)(x-4)$

(Click the arrow to the right to see an example.)

Example 2

Checking The Answer

Question 3: Factoring Quadratic Polynomials 2

(Click the arrow to the right to see the third question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Which of the following ways does $x^2+16$ factor (over the real numbers)?

$(x+4)(x-4)$ $(x-4)^2$ $(x+4)^2$ $x^2+16$ doesn’t factor over the real numbers

(Click the arrow to the right to see an example.)

Example 3 and Checking Our Answer

Advanced Factoring

Question 4: Factoring by Grouping

(Click the arrow to the right to see the fourth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Factor $3x^3+x^2-3x-1$ completely. Please note that this is a different polynomial than the one in the above video.

$(3x+1)(x-1)(x+1)$ $(3x+1)(x^2-1)$ $(3x^2+1)(x+1)$ $(x^2+1)(3x+1)$ You can’t factor polynomials of degree 3 This particular degree 3 polynomial doesn’t factor

(Click the arrow to the right to see an example.)

Example 4

Polynomial Long Division

Introduction and Question 5: Polynomial Long Division

(Click the arrow to the right to see the fifth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

How does $x^5 - 5x^3 + 4x$ factor if either $x=-1$ or $x=1$ is a root of the expression?

$x(x^4-5x^2+4)$ is the furthest the polynomial factors. $x(x-1)(x^4-5x^2+4)$ is the furthest the polynomial factors. $x(x+1)(x^4-5x^2+4)$ is the furthest the polynomial factors. $x(x+1)(x-1)(x+2)(x-2)$ $x(x-1)^2(x+1)(x-4)$ is the furthest the polynomial factors. $x(x^2-1)^2(x^2-4)$ is the furthest the polynomial factors. The polynomial doesn’t factor.

(Click the arrow to the right to see an example.)

Example 5

Synthetic Division

Introduction to Synthetic Division

(Click the arrow to the right an introduction to synthetic division.)

Question 6: Synthetic Long Division

(Click the arrow to the right to see the sixth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Suppose you know that $x+3$ is a factor of the polynomial $f(x) = x^3 - 7x + 6$ . What is the result of dividing $f(x)$ by $x+3$ ?

$x^2-3x+2$ $x^3+3x^2+2x+12$ $x^2 - 7x + 2$ $x^2-5$ $x^2 + 3x + 2$ The result is not a polynomial.

(Click the arrow to the right to see an example.)

Example 6

Question 7: Imaginary Roots

(Click the arrow to the right to see the seventh question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all of the imaginary roots of $x^3 - 3x^2 + 6x - 4$ ?
Hint 1: The graph of the function in the video (when the question is posed) may have some useful information.
Hint 2: If you know one root of the polynomial, synthetic division may be useful.

$x=-1$ , $x=1+\sqrt {3}i$ , and $x= 1-\sqrt {3}i$ $x=1$ , $x=1+\sqrt {3}i$ , and $x= 1-\sqrt {3}i$ $x=-1$ , $x=1+\sqrt {3}i$ , and $x= -1-\sqrt {3}i$ The polynomial has no roots. $x=1$ is the only root of the polynomial $x=-1$ is the only root of the polynomial

(Click the arrow to the right to see an example.)

Example 7

Wrap-up of Factoring Polynomials

Wrap-up