Level 332
Level 334

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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