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Answer :
To multiply the polynomials
[tex]$$ (9x + 8)(3x^2 + x - 1), $$[/tex]
we can use the distributive property (also known as the FOIL method for binomials). Here’s a detailed step-by-step explanation:
1. Distribute the first term of the first polynomial:
Multiply [tex]$9x$[/tex] by each term in the second polynomial:
[tex]\[
9x \cdot 3x^2 = 27x^3,
\][/tex]
[tex]\[
9x \cdot x = 9x^2,
\][/tex]
[tex]\[
9x \cdot (-1) = -9x.
\][/tex]
This gives us:
[tex]\[
27x^3 + 9x^2 - 9x.
\][/tex]
2. Distribute the second term of the first polynomial:
Multiply [tex]$8$[/tex] by each term in the second polynomial:
[tex]\[
8 \cdot 3x^2 = 24x^2,
\][/tex]
[tex]\[
8 \cdot x = 8x,
\][/tex]
[tex]\[
8 \cdot (-1) = -8.
\][/tex]
This results in:
[tex]\[
24x^2 + 8x - 8.
\][/tex]
3. Combine like terms:
Add the two expressions together:
[tex]\[
(27x^3 + 9x^2 - 9x) + (24x^2 + 8x - 8).
\][/tex]
Combine the like terms:
- The cubic term:
[tex]\[
27x^3.
\][/tex]
- The quadratic terms:
[tex]\[
9x^2 + 24x^2 = 33x^2.
\][/tex]
- The linear terms:
[tex]\[
-9x + 8x = -x.
\][/tex]
- The constant term:
[tex]\[
-8.
\][/tex]
Thus, the final result is:
[tex]\[
27x^3 + 33x^2 - x - 8.
\][/tex]
Therefore, the product of [tex]$\left(9x+8\right)$[/tex] and [tex]$\left(3x^2+x-1\right)$[/tex] is:
[tex]$$ \boxed{27x^3+33x^2-x-8}. $$[/tex]
[tex]$$ (9x + 8)(3x^2 + x - 1), $$[/tex]
we can use the distributive property (also known as the FOIL method for binomials). Here’s a detailed step-by-step explanation:
1. Distribute the first term of the first polynomial:
Multiply [tex]$9x$[/tex] by each term in the second polynomial:
[tex]\[
9x \cdot 3x^2 = 27x^3,
\][/tex]
[tex]\[
9x \cdot x = 9x^2,
\][/tex]
[tex]\[
9x \cdot (-1) = -9x.
\][/tex]
This gives us:
[tex]\[
27x^3 + 9x^2 - 9x.
\][/tex]
2. Distribute the second term of the first polynomial:
Multiply [tex]$8$[/tex] by each term in the second polynomial:
[tex]\[
8 \cdot 3x^2 = 24x^2,
\][/tex]
[tex]\[
8 \cdot x = 8x,
\][/tex]
[tex]\[
8 \cdot (-1) = -8.
\][/tex]
This results in:
[tex]\[
24x^2 + 8x - 8.
\][/tex]
3. Combine like terms:
Add the two expressions together:
[tex]\[
(27x^3 + 9x^2 - 9x) + (24x^2 + 8x - 8).
\][/tex]
Combine the like terms:
- The cubic term:
[tex]\[
27x^3.
\][/tex]
- The quadratic terms:
[tex]\[
9x^2 + 24x^2 = 33x^2.
\][/tex]
- The linear terms:
[tex]\[
-9x + 8x = -x.
\][/tex]
- The constant term:
[tex]\[
-8.
\][/tex]
Thus, the final result is:
[tex]\[
27x^3 + 33x^2 - x - 8.
\][/tex]
Therefore, the product of [tex]$\left(9x+8\right)$[/tex] and [tex]$\left(3x^2+x-1\right)$[/tex] is:
[tex]$$ \boxed{27x^3+33x^2-x-8}. $$[/tex]
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