To find the product of [tex]\((4x^3 + 2x^2)(6x - 9)\)[/tex], we will use the distributive property to expand the expression and then combine like terms.
Given:
[tex]\[ (4x^3 + 2x^2)(6x - 9) \][/tex]
Step 1: Distribute each term in the first polynomial to each term in the second polynomial.
- Multiply [tex]\(4x^3\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\[ 4x^3 \cdot 6x = 24x^4 \][/tex]
- Multiply [tex]\(4x^3\)[/tex] by [tex]\(-9\)[/tex]:
[tex]\[ 4x^3 \cdot (-9) = -36x^3 \][/tex]
- Multiply [tex]\(2x^2\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\[ 2x^2 \cdot 6x = 12x^3 \][/tex]
- Multiply [tex]\(2x^2\)[/tex] by [tex]\(-9\)[/tex]:
[tex]\[ 2x^2 \cdot (-9) = -18x^2 \][/tex]
Step 2: Combine the like terms.
- Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[ -36x^3 + 12x^3 = -24x^3 \][/tex]
The final expression is:
[tex]\[ 24x^4 - 24x^3 - 18x^2 \][/tex]
The polynomial after all terms are combined, in descending order of exponents, is:
[tex]\[ 24x^4 - 24x^3 - 18x^2 \][/tex]
