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Level 333

Factoring Polynomials


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8m n^4
24 m^3 n^4 + 32 mn^5 p
5p q^2 r^2
5p^2 q^5 r^2 - 10 p q^2 r^2
4 d^2 e^2
8d^2 e^3 + 12d^3 e^2
(6x + 18)/6
x + 3
(18x +45 x^3)/9x
2 + 5x^2
16x + 4xy - 32xyz
4x(4 + y - 8yz)
12x^4 y^2 z + 42x^3 y^3 z^2
6 x^3 y^2 z (2x + 7yz)
Remainder Theorem
If a polynomial f(x) is divided by (x-c), the remainder is f(c)
Factor Theorem
A polynomial f(x) has a factor (x-c) if and only if f(x) = 0
At most n-1 turning points
A polynomial with degree n has:
A polynomial with odd degree:
Has opposite end behavior on the left and right
A polynomial with even degree:
Has the same end behavior on the left and right
If the multiplicity is odd, it crosses the x-axis at the intercept
When will a polynomial bounce at a zero? When will it cross it?
Intermediate Value Theorem
If f(a) > 0 and f(b) < 0 then there is a real zero between a and b
Rational Zero Theorem
All the rational zeros of a polynomial f(x) will have the form ±p/q where q is a factor of the leading coefficient and p is a factor of the constant term.
IF the leading coefficient of the polynomial is positive:
How do we know if a (positive) number is an upper bound of a polynomial?
How do we know if a (negative) number is a lower bound of a polynomial?
IF the numbers in the bottom row alternate in sign, it is a lower bound
Descartes' Rule of Signs
The number of positive real zeros of f(x) is equal to the number of sign changes in a f(x) decreased by an even integer
Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n>or equal to 1. then P(x)=0 has exactly n roots, including multiple and complex roots.
Linear Factor Theorem
Every polynomial f(x) of degree n can be written as the product of its leading coefficient and n zeros of the form (x-c)
Conjugate Pair Theorem
If a+bi is a zero of a polynomial with real coefficients, a-bi is also a zero.
3x⁵(3 - 4x³ + x⁴)
Use the GCF method to factor: 9x⁵ - 12x⁸ + 3x⁹
5x³y⁴(3x⁵- y²)
Use the GCF method to factor: 15x⁸y⁴- 5x³y⁶
9z(4y⁹- 11z)
Use the GCF method to factor: 36y⁹z - 99z²
(3x + 7)(3x - 7)
Use the Difference of Two Squares method to factor: 9x² - 49
(x + 12)(x - 12)
Use the Difference of Two Squares method to factor: x² - 144
(x + y)(x - y)
Use the Difference of Two Squares method to factor: x² - y²
(6x + 1)(6x - 1)
Use the Difference of Two Squares method to factor: 36x² - 1
(x - 2)(x - 8)
Factor x² - 10x + 16
(x - 6)(x - 7)
Factor x² - 13x + 42
(x - 5)(x + 1)
Factor the trinomial: x² - 4x - 5
(2x + 9)(2x - 9)
Use any method: 4x² - 81
3x(4x² - 3x + 5)
Use any method: 12x³ - 9x² + 15x
5(6x² + 3x + 1)
Use GCF method: 30x² + 15x + 5
(x-5)(x-5)
Use any method: x² - 10x + 25