Question

Factor the trinomial (2x²-11x+14) by expanding the (x)-term using the two factors.

Answer 1

The factored form of the trinomial (2x²-11x+14) is (2x-7)(x-2).

Step-by-step explanation:

To factor the trinomial, we need to find two numbers that multiply to the product of the coefficient of the quadratic term (2) and the constant term (14), which is 28, and add up to the coefficient of the linear term (-11). These numbers are -7 and -4. The factored form is then obtained by splitting the middle term (-11x) using these two numbers:

`\[ 2x^2 - 7x - 4x + 14 \]`

Group the terms and factor out the common factors:

`\[ x(2x - 7) - 2(2x - 7) \]`

Now, notice that we have a common factor of (2x - 7) in both terms. Factor that out:

`\[ (2x - 7)(x - 2) \]`

So, the factored form is (2x - 7)(x - 2). To verify this, you can use the distributive property to multiply these factors back together and simplify to check if it equals the original expression (2x²-11x+14). This factoring method is known as the "AC method," where you find two numbers whose product and sum help in splitting the middle term for factorization.