Factor completely the polynomial. Indicate any that are not factorable using integers.
x 2 – 4 x – 32.
Answer
(x + 4)(x + 8)
(x – 4)(x – 8)
(x + 4)(x – 8)
(x – 4)(x + 8)
not factorable
If I were you, I would expand the possible answers and see what I would get.
Clearly it can't be the first two, since those expansions would give you a + 32 at the end, so ....
To factor completely the polynomial x^2 - 4x - 32, we need to find two binomials that when multiplied together give us the original polynomial. Let's factor it step by step:
1. If we multiply the first terms of both binomials, we need to obtain x^2. This can be achieved by having (x + _ ) and (x - _ ).
2. The constant term -32 can be obtained by multiplying the constants of both binomials.
Now, let's try different combinations to see which one satisfies both conditions:
(x + 4)(x - 8) = x^2 - 8x + 4x - 32 = x^2 - 4x - 32
Therefore, the polynomial can be factored as (x + 4)(x - 8).
To factor the given polynomial, we can use the factoring method. In this case, we have a quadratic trinomial. Here's how we can proceed:
1. First, check if there is a common factor among the coefficients. In this case, we can see that there is no common factor other than 1, so we move on to the next step.
2. We need to find two numbers that multiply to give the constant term (-32) and add up to give the coefficient of the linear term (-4). In this case, the numbers are 8 and -4.
3. Rewrite the quadratic trinomial with the numbers found in step 2, but split the middle term with the two numbers. This gives us:
x^2 - 4x - 32 = x^2 + 8x - 4x - 32
4. Group the terms and factor by grouping:
(x^2 + 8x) + (-4x - 32)
5. From the first group, factor out the greatest common factor, which is x:
x(x + 8) + (-4x - 32)
6. From the second group, factor out the greatest common factor, which is -4:
x(x + 8) - 4(x + 8)
7. Now, we can see that we have a common binomial factor, (x + 8). We can factor it out:
(x + 8)(x - 4)
Therefore, the completely factored form of the polynomial x^2 - 4x - 32 is (x + 8)(x - 4).