Factor $\text{ƒ}\left(x\right)$ into linear factors given that k is a zero. $\text{ƒ}\left(x\right)=x^4+2x^3-7x^2-20x-12;k=-2$ (multiplicity $2$)

Verified step by step guidance

1. Since \( k = -2 \) is a zero of multiplicity 2, it means \( (x + 2)^2 \) is a factor of \( f(x) \). Start by dividing \( f(x) = x^4 + 2x^3 - 7x^2 - 20x - 12 \) by \( (x + 2)^2 \) to find the remaining quadratic factor.
2. First, perform polynomial division or synthetic division to divide \( f(x) \) by \( (x + 2) \) once. This will give you a cubic polynomial as the quotient.
3. Next, divide the cubic polynomial obtained in the previous step by \( (x + 2) \) again, since the multiplicity is 2. This will yield a quadratic polynomial as the quotient.
4. Now, you have factored \( f(x) \) as \( (x + 2)^2 \) times the quadratic polynomial found. The next step is to factor this quadratic polynomial into linear factors, if possible, by using methods such as factoring by inspection, completing the square, or the quadratic formula.
5. Finally, write the complete factorization of \( f(x) \) as the product of \( (x + 2)^2 \) and the linear factors obtained from the quadratic polynomial.

Key Concepts

Polynomial Zeros and Multiplicity

A zero of a polynomial is a value of x that makes the polynomial equal to zero. Multiplicity refers to how many times a particular zero is repeated as a root. For example, if k = -2 is a zero with multiplicity 2, then (x + 2)² is a factor of the polynomial.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a linear factor corresponding to a known zero. Synthetic division is a shortcut method for dividing by linear factors of the form (x - k). This process helps reduce the polynomial's degree and find remaining factors.

Factoring Polynomials into Linear Factors

Factoring a polynomial into linear factors means expressing it as a product of first-degree polynomials. Once zeros and their multiplicities are identified, the polynomial can be written as a product of factors like (x - k)^m, where m is the multiplicity. This fully factors the polynomial over the real or complex numbers.