Factor ƒ(x) into linear factors given that k is a zero. $\text{ƒ}\left(x\right)=-6x^3-25x^2-3x+4;k=-4$

Verified step by step guidance

1. First, recognize that since k = -4 is a zero of the polynomial ƒ(x), it means that (x - (-4)) or (x + 4) is a factor of ƒ(x).
2. Use polynomial division or synthetic division to divide ƒ(x) = -6x^3 - 25x^2 - 3x + 4 by (x + 4). This will give you a quadratic quotient.
3. Set up synthetic division with -4 as the divisor and the coefficients of ƒ(x): -6, -25, -3, and 4.
4. Perform the synthetic division step-by-step to find the quotient polynomial, which will be a quadratic expression.
5. Once you have the quadratic quotient, factor it further into linear factors if possible, so that ƒ(x) is expressed as a product of linear factors including (x + 4).

Key Concepts

Polynomial Zeros and the Factor Theorem

The Factor Theorem states that if k is a zero of a polynomial ƒ(x), then (x - k) is a factor of ƒ(x). This means substituting k into ƒ(x) yields zero, confirming (x - k) divides the polynomial exactly.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide the original polynomial by the factor (x - k) to reduce its degree. Synthetic division is a shortcut method for dividing by linear factors, simplifying the process of factoring higher-degree polynomials.

Factoring Polynomials into Linear Factors

After dividing out the known factor, the resulting polynomial can be further factored into linear factors by finding its zeros. Fully factoring a cubic polynomial means expressing it as a product of three linear factors, if possible.