Let's simplify the given expression step-by-step and classify the resulting polynomial.
Given expression: [tex]\( 3x(x-3) + (2x + 6)(-x - 3) \)[/tex]
1. Simplify each part separately:
- Start with [tex]\( 3x(x - 3) \)[/tex]:
[tex]\[
3x(x - 3) = 3x \cdot x - 3x \cdot 3 = 3x^2 - 9x
\][/tex]
- Next, simplify [tex]\( (2x + 6)(-x - 3) \)[/tex]:
[tex]\[
(2x + 6)(-x - 3)
\][/tex]
Use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(2x + 6)(-x - 3) = 2x \cdot (-x) + 2x \cdot (-3) + 6 \cdot (-x) + 6 \cdot (-3)
\][/tex]
Calculate each term individually:
[tex]\[
2x \cdot (-x) = -2x^2
\][/tex]
[tex]\[
2x \cdot (-3) = -6x
\][/tex]
[tex]\[
6 \cdot (-x) = -6x
\][/tex]
[tex]\[
6 \cdot (-3) = -18
\][/tex]
Combine these terms:
[tex]\[
-2x^2 - 6x - 6x - 18 = -2x^2 - 12x - 18
\][/tex]
2. Combine the simplified parts:
Combine [tex]\( 3x(x-3) \)[/tex] and [tex]\( (2x+6)(-x-3) \)[/tex] results:
[tex]\[
3x^2 - 9x + (-2x^2 - 12x - 18)
\][/tex]
3. Combine like terms:
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
3x^2 - 2x^2 = x^2
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
-9x - 12x = -21x
\][/tex]
- Combine the constant terms:
[tex]\[
-18
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
x^2 - 21x - 18
\][/tex]
4. Classify the polynomial:
The resulting polynomial is [tex]\( x^2 - 21x - 18 \)[/tex].
- Since it has a degree of 2 (the highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex]), it is a quadratic polynomial.
- It has three terms: [tex]\( x^2 \)[/tex], [tex]\(-21x\)[/tex], and [tex]\(-18\)[/tex].
Hence, the resulting polynomial is a quadratic trinomial.
Correct answer: quadratic trinomial
