Sure, let's find the product of the given polynomials step by step:
Given polynomials are:
[tex]\[ P(x) = -3x^5 - 4x^4 \][/tex]
[tex]\[ Q(x) = 7x^2 - 2x + 6 \][/tex]
To find the product [tex]\( P(x) \cdot Q(x) \)[/tex], we distribute each term of [tex]\( P(x) \)[/tex] to each term of [tex]\( Q(x) \)[/tex].
1. Multiply [tex]\(-3x^5\)[/tex] by each term in [tex]\( Q(x) \)[/tex]:
- [tex]\((-3x^5) \cdot (7x^2) = -21x^7\)[/tex]
- [tex]\((-3x^5) \cdot (-2x) = 6x^6\)[/tex]
- [tex]\((-3x^5) \cdot (6) = -18x^5\)[/tex]
2. Multiply [tex]\(-4x^4\)[/tex] by each term in [tex]\( Q(x) \)[/tex]:
- [tex]\((-4x^4) \cdot (7x^2) = -28x^6\)[/tex]
- [tex]\((-4x^4) \cdot (-2x) = 8x^5\)[/tex]
- [tex]\((-4x^4) \cdot (6) = -24x^4\)[/tex]
Next, we combine like terms:
1. Terms involving [tex]\(x^7\)[/tex]:
[tex]\[ -21x^7 \][/tex]
2. Terms involving [tex]\(x^6\)[/tex]:
[tex]\[ 6x^6 + (-28x^6) = -22x^6 \][/tex]
3. Terms involving [tex]\(x^5\)[/tex]:
[tex]\[ -18x^5 + 8x^5 = -10x^5 \][/tex]
4. Terms involving [tex]\(x^4\)[/tex]:
[tex]\[ -24x^4 \][/tex]
Now, let's write the final polynomial by combining all the terms:
[tex]\[
P(x) \cdot Q(x) = -21x^7 - 22x^6 - 10x^5 - 24x^4
\][/tex]
So, the product of the given polynomials is:
[tex]\[
\left(-3x^5 - 4x^4\right)\left(7x^2 - 2x + 6\right) = -21x^7 - 22x^6 - 10x^5 - 24x^4
\][/tex]
