Factor completely: $x^3+x^2-4x-4.$

Verified step by step guidance

1. Group the terms into two pairs: \( (x^3 + x^2) \) and \( (-4x - 4) \). This is called factoring by grouping.
2. Factor out the greatest common factor (GCF) from each pair. From \( (x^3 + x^2) \), the GCF is \( x^2 \), and from \( (-4x - 4) \), the GCF is \( -4 \). This gives \( x^2(x + 1) - 4(x + 1) \).
3. Notice that \( (x + 1) \) is a common factor in both terms. Factor \( (x + 1) \) out, resulting in \( (x + 1)(x^2 - 4) \).
4. Recognize that \( x^2 - 4 \) is a difference of squares. Use the formula \( a^2 - b^2 = (a - b)(a + b) \) to factor \( x^2 - 4 \) into \( (x - 2)(x + 2) \).
5. Combine all the factors to write the fully factored form: \( (x + 1)(x - 2)(x + 2) \).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. Common methods include factoring by grouping, using the distributive property, and applying special product formulas like the difference of squares.

Rational Root Theorem

The Rational Root Theorem provides a way to identify possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem is useful for testing potential roots when factoring.

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, particularly useful when the divisor is a linear polynomial. It allows for quick calculations to determine if a potential root is indeed a root of the polynomial. If the remainder is zero, the divisor is a factor, facilitating the complete factorization of the polynomial.