R.4 Factoring Polynomials

Question 1

What does it mean to factor an expression?

Answer 1

To rewrite the expression as an equivalent product.

Question 2

What is the largest factor common to all terms in a polynomial?

Answer 2

The greatest common factor

Question 3

Can a common binomial factor be factored out of an expression?

Answer 3

Yes

Question 4

What is factoring by grouping?

Answer 4

Grouping the terms using the associative property so that terms can be factored down to show a common binomial factor, which is then factored out.

Question 5

What end result are we trying to arrive at?

Answer 5

A completely factored form

Question 6

What is the first thing to look for when factoring?

Answer 6

Common factors

Question 7

What is the form of a quadratic polynomial?

Answer 7

ax^2 + bx + c

Where a, b, c are real numbers, and a != 0

Question 8

How do you factor a quadratic expression where a = 1, i.e. x^2 + bx + c?

Answer 8

The first term in each binomial will be x:
(x ) (x )

The last terms will have a product that equals c in the quadratic.

The middle terms will have a sum that equals b in the quadratic.

Example: x^2 - 8x + 7
(x-7) (x-1)

Question 9

How can you tell what the signs will be inside the binomial expressions that result from factoring a quadratic?

Answer 9

If the constant term is positive, the binomial expressions will have like signs.

If the constant term is negative, the binomial will have unlike signs.

If the signs are like, the sign of the linear term will match the signs of the binomial terms.

Question 10

What is the term for a polynomial that cannot be factored?

Answer 10

A prime polynomial

Question 11

How do you factor a quadratic expression where the leading coefficient is not 1?

Answer 11

Trial-and-error

Question 12

How do you factor the difference of two squares?

Answer 12

Given any expression that can be written:
A^2 - B^2:

(A + B) (A - B)

Question 13

How is the sum of two perfect squares factored?

Answer 13

It cannot be! The sum of two perfect squares is prime.

Question 14

What does a perfect square trinomial factor into?

Answer 14

A perfect square.

Question 15

How do you identify a perfect square trinomial?

Answer 15

The first and last terms are perfect squares and the middle term is twice the product of the two terms.

Example: x^2 + 14x + 49

Question 16

How do you factor the sum or difference of two perfect cubes?

Answer 16

1) Each will factor into the product of a binomial and a trinomial.
2) The terms of the binomial are the quantities being cubed.
3) The terms of the trinomial are the square of A, the product of AB, and the square of B, respectively.
4) The binomial takes the same sign as the original expression.
5) The middle term of the trinomial takes the opposite sign of the original exercise. The last term is always positive.

Question 17

How do you substitute using “u”?

Answer 17

This can be used when you have expressions in quadratic form.

Because the quadratic form is always Ax^2 + Bx + C, you can substitute the variable “u” for x. This simplifies the solving process when x is complicated. At the end of the factoring, just plug x back in for “u”.