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The problem asks to factor the quadratic expression $6x^2 - 11x + 5$, if possible. If it cannot be factored, then the polynomial is prime.

AlgebraQuadratic EquationsFactorizationPolynomials
2025/3/25

1. Problem Description

The problem asks to factor the quadratic expression 6x2−11x+56x^2 - 11x + 5, if possible. If it cannot be factored, then the polynomial is prime.

2. Solution Steps

We need to factor the quadratic expression 6x2−11x+56x^2 - 11x + 5. We are looking for two binomials of the form (ax+b)(cx+d)(ax + b)(cx + d) such that ac=6ac = 6, bd=5bd = 5, and ad+bc=−11ad + bc = -11.
We can list possible factors of 6 and 5:
Factors of 6: 1 and 6, 2 and 3
Factors of 5: 1 and 5
Let's try different combinations:
(x−1)(6x−5)=6x2−5x−6x+5=6x2−11x+5(x - 1)(6x - 5) = 6x^2 - 5x - 6x + 5 = 6x^2 - 11x + 5.
This factorization works.
Alternatively,
(2x−5)(3x−1)=6x2−2x−15x+5=6x2−17x+5(2x - 5)(3x - 1) = 6x^2 - 2x - 15x + 5 = 6x^2 - 17x + 5.
This does not work.
(2x−1)(3x−5)=6x2−10x−3x+5=6x2−13x+5(2x - 1)(3x - 5) = 6x^2 - 10x - 3x + 5 = 6x^2 - 13x + 5.
This does not work.
Thus, the factorization is (x−1)(6x−5)(x - 1)(6x - 5).

3. Final Answer

6x2−11x+5=(x−1)(6x−5)6x^2 - 11x + 5 = (x-1)(6x-5)

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