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Answer :
To multiply the polynomials
[tex]$$\left(3x^2 - 4x + 5\right) \quad \text{and} \quad \left(x^2 - 3x + 2\right),$$[/tex]
we use the distributive property, multiplying each term in the first polynomial by every term in the second polynomial.
1. Multiply the first term in the first polynomial by each term in the second polynomial:
[tex]$$
\begin{aligned}
3x^2 \cdot x^2 &= 3x^4, \\
3x^2 \cdot (-3x) &= -9x^3, \\
3x^2 \cdot 2 &= 6x^2.
\end{aligned}
$$[/tex]
2. Multiply the second term in the first polynomial by each term in the second polynomial:
[tex]$$
\begin{aligned}
-4x \cdot x^2 &= -4x^3, \\
-4x \cdot (-3x) &= 12x^2, \\
-4x \cdot 2 &= -8x.
\end{aligned}
$$[/tex]
3. Multiply the third term in the first polynomial by each term in the second polynomial:
[tex]$$
\begin{aligned}
5 \cdot x^2 &= 5x^2, \\
5 \cdot (-3x) &= -15x, \\
5 \cdot 2 &= 10.
\end{aligned}
$$[/tex]
Now, combine the like terms:
- For the [tex]$x^4$[/tex] term:
[tex]$$
3x^4.
$$[/tex]
- For the [tex]$x^3$[/tex] terms:
[tex]$$
-9x^3 - 4x^3 = -13x^3.
$$[/tex]
- For the [tex]$x^2$[/tex] terms:
[tex]$$
6x^2 + 12x^2 + 5x^2 = 23x^2.
$$[/tex]
- For the [tex]$x$[/tex] terms:
[tex]$$
-8x - 15x = -23x.
$$[/tex]
- For the constant term:
[tex]$$
10.
$$[/tex]
Thus, the product of the two polynomials is:
[tex]$$
3x^4 - 13x^3 + 23x^2 - 23x + 10.
$$[/tex]
Comparing this with the provided choices:
- A. [tex]$4x^2 - 7x + 7$[/tex]
- B. [tex]$3x^4 + 12x^2 + 10$[/tex]
- C. [tex]$3x^4 - 13x^3 + 23x^2 - 23x + 10$[/tex]
- D. [tex]$3x^4 + 10x^2 + 12x + 10$[/tex]
The correct answer is Option C.
[tex]$$\left(3x^2 - 4x + 5\right) \quad \text{and} \quad \left(x^2 - 3x + 2\right),$$[/tex]
we use the distributive property, multiplying each term in the first polynomial by every term in the second polynomial.
1. Multiply the first term in the first polynomial by each term in the second polynomial:
[tex]$$
\begin{aligned}
3x^2 \cdot x^2 &= 3x^4, \\
3x^2 \cdot (-3x) &= -9x^3, \\
3x^2 \cdot 2 &= 6x^2.
\end{aligned}
$$[/tex]
2. Multiply the second term in the first polynomial by each term in the second polynomial:
[tex]$$
\begin{aligned}
-4x \cdot x^2 &= -4x^3, \\
-4x \cdot (-3x) &= 12x^2, \\
-4x \cdot 2 &= -8x.
\end{aligned}
$$[/tex]
3. Multiply the third term in the first polynomial by each term in the second polynomial:
[tex]$$
\begin{aligned}
5 \cdot x^2 &= 5x^2, \\
5 \cdot (-3x) &= -15x, \\
5 \cdot 2 &= 10.
\end{aligned}
$$[/tex]
Now, combine the like terms:
- For the [tex]$x^4$[/tex] term:
[tex]$$
3x^4.
$$[/tex]
- For the [tex]$x^3$[/tex] terms:
[tex]$$
-9x^3 - 4x^3 = -13x^3.
$$[/tex]
- For the [tex]$x^2$[/tex] terms:
[tex]$$
6x^2 + 12x^2 + 5x^2 = 23x^2.
$$[/tex]
- For the [tex]$x$[/tex] terms:
[tex]$$
-8x - 15x = -23x.
$$[/tex]
- For the constant term:
[tex]$$
10.
$$[/tex]
Thus, the product of the two polynomials is:
[tex]$$
3x^4 - 13x^3 + 23x^2 - 23x + 10.
$$[/tex]
Comparing this with the provided choices:
- A. [tex]$4x^2 - 7x + 7$[/tex]
- B. [tex]$3x^4 + 12x^2 + 10$[/tex]
- C. [tex]$3x^4 - 13x^3 + 23x^2 - 23x + 10$[/tex]
- D. [tex]$3x^4 + 10x^2 + 12x + 10$[/tex]
The correct answer is Option C.
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