How do you factor X^3+5x^2-28x-32?
pls help
let f(x) = x^3 + 5x^2 - 28x - 32
I tried x = ±1, ±2, ±4 (that is, factors of 32)
on the second try, I found f(-1) = 0
so x+1) is a factor
Using synthetic division I found
x^3 + 5x^2 - 28x - 32 = (x+1)(x^2 + 4x -32) which factored again to
(x+1)(x+8)(x-4)
To factor the polynomial X^3 + 5x^2 - 28x - 32, we can use a method called synthetic division.
Step 1: We begin by looking for possible rational roots of the polynomial. The rational roots can be found by taking the factors of the constant term (-32 in this case) and dividing them by the factors of the leading coefficient (1 in this case).
The factors of -32 are ±1, ±2, ±4, ±8, ±16, and ±32. The factors of 1 are ±1.
Step 2: We test these possible rational roots using synthetic division. Choose one of the possible roots and divide the polynomial by it. If the remainder is 0, then the chosen root is a factor of the polynomial. Repeat this process until all possible rational roots have been tested.
Let's start with the possible root x = 1.
1 │ 1 5 -28 -32
────
1 6 -22 -10
The result of one step of synthetic division gives us a remainder of -10, which means 1 is not a factor.
Now let's try x = -1.
-1 │ 1 5 -28 -32
────
-1 -4 32
─────
1 1 4
The result of this synthetic division gives us a remainder of 4, which means -1 is not a factor.
Let's try x = 2.
2 │ 1 5 -28 -32
────
2 14 -28
───────
1 7 -6
The result of this synthetic division gives us a remainder of -6, which means 2 is not a factor.
Finally, let's try x = -2.
-2 │ 1 5 -28 -32
────
-2 -6 68
──────
1 3 36
─────────
1 4 0
The result of this synthetic division gives us a remainder of 0, which means -2 is a factor of the polynomial.
Step 3: Since -2 is a factor, we can rewrite the polynomial as a product:
X^3 + 5x^2 - 28x - 32 = (x + 2)(x^2 + 3x + 16)
Now, we need to factor the quadratic expression x^2 + 3x + 16.
Unfortunately, this quadratic cannot be factored using real numbers since its discriminant (b^2 - 4ac) is negative.
So the final factored form of the polynomial X^3 + 5x^2 - 28x - 32 is:
(X + 2)(x^2 + 3x + 16)