Which of the following is a factor of $10 x^2-11 x-6$?
The goal is to find which of the given expressions is a factor of the quadratic polynomial \(10x^2 - 11x - 6\). To do this, we factor the quadratic polynomial. We can use the AC method (also known as the grouping method), which involves looking for two numbers that multiply to the product of the leading coefficient and the constant term, and add up to the middle coefficient. Here, the leading coefficient is 10 and the constant term is -6, so the product is \(10 \times (-6) = -60\). The middle coefficient is -11. We look for two integers whose product is -60 and whose sum is -11. Checking the factors of -60, we find that -15 and 4 satisfy these conditions: \(-15 \times 4 = -60\) and \(-15 + 4 = -11\).
Now we use these numbers to rewrite the middle term of the original expression:
\[10x^2 - 11x - 6 = 10x^2 - 15x + 4x - 6.\]
Next, we group the terms:
\[(10x^2 - 15x) + (4x - 6).\]
Now, we extract the greatest common factor from each group. From the first group, we can factor out \(5x\), and from the second group, we can factor out \(2\):
\[5x(2x - 3) + 2(2x - 3).\]
Since both terms contain the binomial \((2x - 3)\), we can factor it out:
\[(5x + 2)(2x - 3).\]
The factors of the polynomial \(10x^2 - 11x - 6\) are \((5x + 2)\) and \((2x - 3)\).
We compare our result with the provided options:
Therefore, the factor that appears in the options is \(5x + 2\). The correct choice is the second option.
Factoring polynomials is essential for solving quadratic equations, which frequently arise in real-life problems like determining the trajectory of a projectile or finding the area of a shape. For example, if you are designing a rectangular garden with an area given by the expression \(10x^2 - 11x - 6\) square meters, you could factor this expression to find that the dimensions of the garden are \((5x + 2)\) meters and \((2x - 3)\) meters. Being able to break down complex expressions into simpler factors makes these real-world problems much easier to model and solve!