Question

Which expression is a factor of x^3-8x^2+x+42 ?

Answer 1

To determine a factor of the polynomial \(x^3 - 8x^2 + x + 42\), we can use the Rational Root Theorem. This theorem suggests that any rational root, in the form of \(\frac{p}{q}\), must have \(p\) as a factor of the constant term (42) and \(q\) as a factor of the leading coefficient (1). Since the leading coefficient is 1, we will look for integer factors.

The integer factors of 42 are:
\[
\pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42
\]

We will test these factors to see if any are roots of the polynomial. We can substitute these values into the polynomial \(f(x) = x^3 - 8x^2 + x + 42\) and check if it equals zero.

Let's try \(x = 2\):
\[
f(2) = 2^3 - 8 \cdot 2^2 + 2 + 42
\]
\[
= 8 - 32 + 2 + 42
\]
\[
= 8 - 32 + 2 + 42 = 20 \quad (\text{not a root})
\]

Next, let's try \(x = -1\):
\[
f(-1) = (-1)^3 - 8(-1)^2 + (-1) + 42
\]
\[
= -1 - 8 - 1 + 42
\]
\[
= -1 - 8 - 1 + 42 = 32 \quad (\text{not a root})
\]

Now let’s try \(x = 3\):
\[
f(3) = 3^3 - 8 \cdot 3^2 + 3 + 42
\]
\[
= 27 - 72 + 3 + 42
\]
\[
= 27 - 72 + 3 + 42 = 0 \quad (\text{is a root})
\]

Since \(x = 3\) is a root, we can use polynomial long division to factor \(x^3 - 8x^2 + x + 42\) by \(x - 3\).

Performing the division, we have:

1. Divide the leading term \(x^3\) by \(x\) (which is \(x^2\)).
2. Multiply \(x^2\) by \(x - 3\) to get \(x^3 - 3x^2\).
3. Subtract this product from the original polynomial:
\[ (x^3 - 8x^2) - (x^3 - 3x^2) = -5x^2 \]
4. Bring down the next term (this gives us \(-5x^2 + x\)).
5. Divide \(-5x^2\) by \(x\) (which is \(-5x\)).
6. Multiply \(-5x\) by \(x - 3\) to get \(-5x^2 + 15x\).
7. Subtract:
\[ (-5x^2 + x) - (-5x^2 + 15x) = -14x \]
8. Bring down the last term (this gives \(-14x + 42\)).
9. Divide \(-14x\) by \(x\) (which is \(-14\)).
10. Multiply \(-14\) by \(x - 3\) to get \(-14x + 42\) and subtract:
\[ (-14x + 42) - (-14x + 42) = 0 \]

Thus, we have:
\[
x^3 - 8x^2 + x + 42 = (x - 3)(x^2 - 5x - 14)
\]

Now, the quadratic \(x^2 - 5x - 14\) can also be factored or solved using the quadratic formula, but the factorization we are looking for right now is:

The expression \(x - 3\) is therefore a factor of \(x^3 - 8x^2 + x + 42\).