Factor ƒ(x) into linear factors given that k is a zero. $\text{ƒ}\left(x\right)=2x^3-3x^2-5x+6{;}^{}k=1$

Verified step by step guidance

1. Since k = 1 is a zero of the polynomial ƒ(x) = 2x^3 - 3x^2 - 5x + 6, use synthetic division or polynomial division to divide ƒ(x) by (x - 1).
2. Set up synthetic division with the coefficients of ƒ(x): 2, -3, -5, and 6, and use 1 as the divisor.
3. Perform the synthetic division step-by-step to find the quotient polynomial, which will be a quadratic expression.
4. Write ƒ(x) as the product of (x - 1) and the quotient polynomial obtained from the division.
5. Factor the quadratic quotient further, if possible, into linear factors to express ƒ(x) completely as a product of linear factors.

Key Concepts

Polynomial Zeros and the Factor Theorem

A zero of a polynomial is a value of x that makes the polynomial equal to zero. The Factor Theorem states that if k is a zero of ƒ(x), then (x - k) is a factor of ƒ(x). This allows us to factor the polynomial by dividing it by (x - k).

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a linear factor like (x - k). Synthetic division is a shortcut method for dividing by linear factors and helps find the quotient polynomial after factoring out (x - k). This simplifies the polynomial into lower-degree factors.

Factoring Polynomials into Linear Factors

After dividing out a known factor, the resulting polynomial can often be factored further into linear factors by finding its zeros. Factoring completely into linear factors means expressing the polynomial as a product of first-degree polynomials, which corresponds to all its roots.