Factoring Polynomials

The process of factoring polynomials is essentially deconstructing a polynomial into its most basic elements, known as factors. These factors, when multiplied, will yield the original polynomial. This process bears considerable similarity to the prime factorization of numerical values. It is a vital tool in simplifying intricate expressions, providing solutions to polynomial equations, and assessing the properties of polynomial functions.

Finding the GCF of a Polynomial

Find the greatest common factor (GCF) of the polynomial \(12x^3 + 16x^2 + 20x\).

Factoring Out Greatest Common Factor (GCF)

Factor out the greatest common factor (GCF) from the polynomial \(12x^3 - 36x^2 + 24x\).

Identifying the Common Factors

Factor the polynomial \(12x^3 - 36x^2 + 24x\)

Cancelling the Common Factors

Simplify the expression \(\frac{x^3 - 3x^2 + 2x}{x^2 - x}\)

Factoring Trinomials

Factor the trinomial \(x^2 - 3x - 4\).

Trinomial Squares

Factorize the following trinomial square: \(x^2 - 6x + 9\)

Factoring Using Any Method

Factorize the polynomial \(2x^3 - 5x^2 - 23x + 30\).

Factoring a Difference of Squares

Factorize the following polynomial using the difference of squares method: \(x^2 - 9\)

Factoring a Sum of Cubes

Factorize the sum of cubes \(x^3 + 8\)

Factoring by Grouping

Factorize the polynomial \(3x^3 - 6x^2 + 2x - 4\) by grouping.

Factoring a Difference of Cubes

Factor the polynomial \(x^3 - 8\)

Determine if an Expression is a Factor

Is \(x+2\) a factor of the polynomial \(2x^2 + 4x + 2\)?

Determining if Factor Using Synthetic Division

Determine if the polynomial \(4x^4 - 10x^3 + 4x^2 - 10x + 4\) can be factored by the binomial \(2x-1\) using synthetic division.

Find the Factors Using the Factor Theorem

Find the complex zeros of the following polynomial function. Write $f$ in factored form. \[ f(x)=x^{3}-8 x^{2}+25 x-26 \]

Determining if the Polynomial is a Perfect Square

Determine if the polynomial \(x^4 - 2x^2y^2 + y^4\) is a perfect square. If so, factor it.

Expand using the Binomial Theorem

Expand the binomial \((x+2)^4\) using the Binomial Theorem, and then factor the resulting polynomial.

Finding All Integers k Such That the Trinomial Can Be Factored

Find all integers \( k \) such that the trinomial \( x^2 - kx + 12 \) can be factored over the integers.