To factor the given polynomial, [tex]\(-0.9 x^2 - 6.3 x + 16.2\)[/tex], we need to rewrite it in a factored form. Let's go through the process step-by-step.
1. Identify the polynomial: We start with the expression:
[tex]\[
-0.9 x^2 - 6.3 x + 16.2
\][/tex]
2. Factor out the common factor: Notice that [tex]\(-0.9\)[/tex] is a common factor for each term in the polynomial. Factoring [tex]\(-0.9\)[/tex] out, we have:
[tex]\[
-0.9 (x^2 + 7 x - 18)
\][/tex]
Here, you can see that [tex]\(a = -0.9\)[/tex], [tex]\(b = -6.3\)[/tex], and [tex]\(c = 16.2\)[/tex]. Factoring out [tex]\(-0.9\)[/tex] simplifies the polynomial.
3. Focus on the quadratic expression: Now we need to factor the quadratic expression inside the parentheses:
[tex]\[
x^2 + 7 x - 18
\][/tex]
4. Look for two numbers that multiply to the constant term (-18) and add to the coefficient of the [tex]\(x\)[/tex] term (7):
By trial, error, or use of factoring techniques, we find that these numbers are 9 and -2.
5. Write the quadratic expression as a product of binomials:
[tex]\[
x^2 + 7 x - 18 = (x + 9)(x - 2)
\][/tex]
6. Combine this factored form back with the common factor:
[tex]\[
-0.9 (x^2 + 7 x - 18) = -0.9 (x + 9)(x - 2)
\][/tex]
7. Verify the result by expanding:
To confirm our factorization, we can expand [tex]\((-0.9 (x + 9)(x - 2))\)[/tex]:
[tex]\[
(x + 9)(x - 2) = x^2 - 2x + 9x - 18 = x^2 + 7x - 18
\][/tex]
This confirms that:
[tex]\[
-0.9 (x + 9)(x - 2) = -0.9 x^2 - 6.3 x + 16.2
\][/tex]
Hence, the polynomial [tex]\(-0.9 x^2 - 6.3 x + 16.2\)[/tex] can be factored as:
[tex]\[
\boxed{-0.9 (x + 9)(x - 2)}
\][/tex]
Thus, the correct answer is:
A. [tex]\(-0.9 (x + 9)(x - 2)\)[/tex]
