Factor this polynomial completely.10x2 – 11x + 3A.(2x – 1)(5x – 3)B.(5x – 3)(x – 1)C.(5x – 1)(x – 3)D.(2x – 3)(5x – 1)

1. Break Down the Problem

We need to factor the polynomial $10x^2 - 11x + 3$ completely.

2. Relevant Concepts

To factor a quadratic polynomial of the form $ax^2 + bx + c$, we can use the method of finding two numbers that multiply to $ac$ (where $a = 10$ and $c = 3$) and add to $b$ (where $b = -11$).

3. Analysis and Detail

1. Calculate $ac$: $ac = 10 \times 3 = 30$

2. Find two numbers that multiply to $30$ and add to $-11$. The numbers are $-6$ and $-5$ because: $-6 \times -5 = 30 \quad \text{and} \quad -6 + -5 = -11$

3. Rewrite the middle term using the found numbers: $10x^2 - 6x - 5x + 3$

4. Factor by grouping:

   • Group: $(10x^2 - 6x) + (-5x + 3)$
   • Factor out common factors: $2x(5x - 3) -1(5x - 3)$
   • This can be factored further: $(2x - 1)(5x - 3)$

4. Verify and Summarize

Now we have $10x^2 - 11x + 3 = (2x - 1)(5x - 3)$.

Let's verify:

1. Expand $(2x - 1)(5x - 3)$: $= 2x \cdot 5x + 2x \cdot (-3) - 1 \cdot 5x - 1 \cdot (-3) = 10x^2 - 6x - 5x + 3 = 10x^2 - 11x + 3$

The factorization is correct.

Final Answer

The completely factored form of the polynomial $10x^2 - 11x + 3$ is $(2x - 1)(5x - 3)$, which corresponds to option A. (2x – 1)(5x – 3).