R.4 Factoring Polynomials Flashcards
What does it mean to factor an expression?
To rewrite the expression as an equivalent product.
What is the largest factor common to all terms in a polynomial?
The greatest common factor
Can a common binomial factor be factored out of an expression?
Yes
What is factoring by grouping?
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.
What end result are we trying to arrive at?
A completely factored form
What is the first thing to look for when factoring?
Common factors
What is the form of a quadratic polynomial?
ax^2 + bx + c
Where a, b, c are real numbers, and a != 0
How do you factor a quadratic expression where a = 1, i.e. x^2 + bx + c?
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)
How can you tell what the signs will be inside the binomial expressions that result from factoring a quadratic?
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.
What is the term for a polynomial that cannot be factored?
A prime polynomial
How do you factor a quadratic expression where the leading coefficient is not 1?
Trial-and-error
How do you factor the difference of two squares?
Given any expression that can be written:
A^2 - B^2:
(A + B) (A - B)
How is the sum of two perfect squares factored?
It cannot be! The sum of two perfect squares is prime.
What does a perfect square trinomial factor into?
A perfect square.
How do you identify a perfect square trinomial?
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
How do you factor the sum or difference of two perfect cubes?
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.
How do you substitute using “u”?
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”.
