Question

Write the polynomial x² - 12x + 32 as the product of two first degree polynomials.

Hint:

When factoring a quadratic x² + bx + c, pay attention to the signs. If c is positive, the two numbers have the same sign (both positive if b is positive, both negative if b is negative). If c is negative, the two numbers have opposite signs.

Answer 1

We are asked to factorize the given quadratic polynomial. A "first degree polynomial" is a linear expression of the form ax+b.

To factor a quadratic of the form x² + bx + c, we need to find two numbers that multiply to give c and add to give b.

The given polynomial is x² - 12x + 32.
Here, b = -12 and c = 32.
We need to find two numbers that:
- Multiply to 32
- Add up to -12
Let's list pairs of factors of 32: (1, 32), (2, 16), (4, 8).
Since the sum is negative (-12) and the product is positive (32), both numbers must be negative.
Let's check the pairs with negative signs:
- (-1) + (-32) = -33
- (-2) + (-16) = -18
- (-4) + (-8) = -12
The pair -4 and -8 satisfies both conditions.
Therefore, the polynomial can be factored as (x - 4)(x - 8).

The polynomial as a product of two first-degree polynomials is (x-4)(x-8).