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Answer :
To multiply the polynomials
[tex]$$
(3x^2 - 4x + 5) \quad \text{and} \quad (x^2 - 3x + 2),
$$[/tex]
we distribute each term of the first polynomial with each term of the second polynomial.
Step 1. Multiply [tex]$3x^2$[/tex] by each term of the second polynomial:
- [tex]$3x^2 \cdot x^2 = 3x^4$[/tex]
- [tex]$3x^2 \cdot (-3x) = -9x^3$[/tex]
- [tex]$3x^2 \cdot 2 = 6x^2$[/tex]
Step 2. Multiply [tex]$-4x$[/tex] by each term of the second polynomial:
- [tex]$-4x \cdot x^2 = -4x^3$[/tex]
- [tex]$-4x \cdot (-3x) = 12x^2$[/tex]
- [tex]$-4x \cdot 2 = -8x$[/tex]
Step 3. Multiply [tex]$5$[/tex] by each term of the second polynomial:
- [tex]$5 \cdot x^2 = 5x^2$[/tex]
- [tex]$5 \cdot (-3x) = -15x$[/tex]
- [tex]$5 \cdot 2 = 10$[/tex]
Step 4. Combine like terms:
- The only [tex]$x^4$[/tex] term is:
[tex]$$
3x^4.
$$[/tex]
- Combine the [tex]$x^3$[/tex] terms:
[tex]$$
-9x^3 + (-4x^3) = -13x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
6x^2 + 12x^2 + 5x^2 = 23x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
-8x + (-15x) = -23x.
$$[/tex]
- The constant term is:
[tex]$$
10.
$$[/tex]
Thus, the product is:
[tex]$$
3x^4 - 13x^3 + 23x^2 - 23x + 10.
$$[/tex]
Comparing with the given options, this corresponds to option B.
[tex]$$
(3x^2 - 4x + 5) \quad \text{and} \quad (x^2 - 3x + 2),
$$[/tex]
we distribute each term of the first polynomial with each term of the second polynomial.
Step 1. Multiply [tex]$3x^2$[/tex] by each term of the second polynomial:
- [tex]$3x^2 \cdot x^2 = 3x^4$[/tex]
- [tex]$3x^2 \cdot (-3x) = -9x^3$[/tex]
- [tex]$3x^2 \cdot 2 = 6x^2$[/tex]
Step 2. Multiply [tex]$-4x$[/tex] by each term of the second polynomial:
- [tex]$-4x \cdot x^2 = -4x^3$[/tex]
- [tex]$-4x \cdot (-3x) = 12x^2$[/tex]
- [tex]$-4x \cdot 2 = -8x$[/tex]
Step 3. Multiply [tex]$5$[/tex] by each term of the second polynomial:
- [tex]$5 \cdot x^2 = 5x^2$[/tex]
- [tex]$5 \cdot (-3x) = -15x$[/tex]
- [tex]$5 \cdot 2 = 10$[/tex]
Step 4. Combine like terms:
- The only [tex]$x^4$[/tex] term is:
[tex]$$
3x^4.
$$[/tex]
- Combine the [tex]$x^3$[/tex] terms:
[tex]$$
-9x^3 + (-4x^3) = -13x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
6x^2 + 12x^2 + 5x^2 = 23x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
-8x + (-15x) = -23x.
$$[/tex]
- The constant term is:
[tex]$$
10.
$$[/tex]
Thus, the product is:
[tex]$$
3x^4 - 13x^3 + 23x^2 - 23x + 10.
$$[/tex]
Comparing with the given options, this corresponds to option B.
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