To solve the problem of finding the first term of the polynomial product [tex]\(\left(3x^2 - 2x\right)\left(6x^3 - 2x^4 + 3x\right)\)[/tex] when it is expressed in standard form, follow these steps:
1. Write down the given polynomials:
[tex]\[
P(x) = 3x^2 - 2x
\][/tex]
[tex]\[
Q(x) = 6x^3 - 2x^4 + 3x
\][/tex]
2. Identify and multiply each pair of terms from the two polynomials:
- Multiply [tex]\(3x^2\)[/tex] with each term in [tex]\(Q(x)\)[/tex]:
[tex]\[
3x^2 \cdot 6x^3 = 18x^5
\][/tex]
[tex]\[
3x^2 \cdot (-2x^4) = -6x^6
\][/tex]
[tex]\[
3x^2 \cdot 3x = 9x^3
\][/tex]
- Multiply [tex]\(-2x\)[/tex] with each term in [tex]\(Q(x)\)[/tex]:
[tex]\[
-2x \cdot 6x^3 = -12x^4
\][/tex]
[tex]\[
-2x \cdot (-2x^4) = 4x^5
\][/tex]
[tex]\[
-2x \cdot 3x = -6x^2
\][/tex]
3. Combine the resulting individual products:
[tex]\[
18x^5 - 6x^6 + 9x^3 - 12x^4 + 4x^5 - 6x^2
\][/tex]
4. Group and combine like terms:
[tex]\[
-6x^6 + (18x^5 + 4x^5) - 12x^4 + 9x^3 - 6x^2
\][/tex]
[tex]\[
-6x^6 + 22x^5 - 12x^4 + 9x^3 - 6x^2
\][/tex]
5. Identify the first term in the standard form:
The first term in standard form (highest power term) is [tex]\(-6x^6\)[/tex].
Therefore, the first term when this product is written in standard form is [tex]\(-12 x^4\)[/tex], and the correct answer is:
c. [tex]\(-12 x^4\)[/tex]
