Notes
- Factoring Polynomials
- We already learned how to factor quadratic equations, but to factor different types of polynomials, we need more methods, in addition to the ones we already know.
- Whenever you recognize a sum of two cubes, it will always factor into the pattern (a3 + b3) = (a + b)(a2 - ab + b2). Do not worry about signs; they are "built-in" to the equation.
- The difference of two cubes will always factor into the pattern (a3 - b3) = (a - b)(a2 + ab + b2). Do not worry about signs; they are "built-in" to the equation.
- Another way to factor polynomials is by grouping.
- A polynomial that will factor by grouping will factor into the pattern (fa + fb + sa + sb) = f(a + b) + s(a + b) = (f + s)(a + b). The only way to know if a polynomial factors by grouping is to check and see if you can take a factor out of two terms and a different factor out of the other two terms and have the same thing left.
- Other polynomials that resemble quadratic equations can be factored using the same methods used to factor quadratic equations, like double-bubbling ("un/de-foiling"), or difference of squares.
- Solving Polynomial Equations
- If you can factor polynomial expressions, solving polynomial equations is easy.
- Just write the polynomial part in standard form and set it equal to zero if it is not already.
- Then, factor the polynomial completely.
- Lastly, use the zero product property (the one that lets you set the factored parts equal to zero) to solve.
- There are as many solutions as the degree of the polynomial, although some may be irrational or even imaginary.
Practice Quiz
Factor completely.
- 12x3 + 6x2 - 4x - 1
- x3 - x2 + x - 1
- x4 - 9x2 + 20
- 27x3 + 64y3
- 8 - x3
- 27x3 + 216 = 0
- 16x4 - 24x2 + 9 = 0
Answers
- (6x2 - 4)(x + 1)
- (x - 1)(x2 + 1)
- (x2 - 5)(x + 2)(x - 2)
- (3x + 4x)(9x2 - 12xy + 16y2)
- (2 - x)(4 + 2x + x2)
- -8
(2) / 4
Sections
Section 1 - Using Properties of Exponents
Section 2 - Evaluating & Graphing Polynomial Functions
Section 3 - Adding, Subtracting, & Multiplying Polynomials
Section 4 - Factoring & Solving Polynomial Equations
Section 5 - The Remainder & Factor Theorems
Section 6 - Finding Rational Zeroes
Section 7 - Using the Fundamental Theorem of Algebra
Chapter 6 Review Home