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Answer :
To multiply the polynomials
[tex]$$
(7x^2 + 9x + 7)(9x - 4),
$$[/tex]
we distribute each term in the first polynomial to each term in the second polynomial. Here is a step-by-step explanation:
1. Multiply the first term of the first polynomial, [tex]$7x^2$[/tex], by each term in the second polynomial:
[tex]$$
7x^2 \cdot 9x = 63x^3 \quad \text{and} \quad 7x^2 \cdot (-4) = -28x^2.
$$[/tex]
So, from the term [tex]$7x^2$[/tex] we obtain:
[tex]$$
63x^3 - 28x^2.
$$[/tex]
2. Multiply the second term, [tex]$9x$[/tex], by each term in the second polynomial:
[tex]$$
9x \cdot 9x = 81x^2 \quad \text{and} \quad 9x \cdot (-4) = -36x.
$$[/tex]
This gives:
[tex]$$
81x^2 - 36x.
$$[/tex]
3. Multiply the third term, [tex]$7$[/tex], by each term in the second polynomial:
[tex]$$
7 \cdot 9x = 63x \quad \text{and} \quad 7 \cdot (-4) = -28.
$$[/tex]
From this, we get:
[tex]$$
63x - 28.
$$[/tex]
4. Now, combine all the resulting terms:
[tex]$$
63x^3 - 28x^2 + 81x^2 - 36x + 63x - 28.
$$[/tex]
5. Combine like terms:
- The [tex]$x^3$[/tex] term is:
[tex]$$
63x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
-28x^2 + 81x^2 = 53x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
-36x + 63x = 27x.
$$[/tex]
- The constant is:
[tex]$$
-28.
$$[/tex]
Thus, the product of the polynomials is:
[tex]$$
63x^3 + 53x^2 + 27x - 28.
$$[/tex]
[tex]$$
(7x^2 + 9x + 7)(9x - 4),
$$[/tex]
we distribute each term in the first polynomial to each term in the second polynomial. Here is a step-by-step explanation:
1. Multiply the first term of the first polynomial, [tex]$7x^2$[/tex], by each term in the second polynomial:
[tex]$$
7x^2 \cdot 9x = 63x^3 \quad \text{and} \quad 7x^2 \cdot (-4) = -28x^2.
$$[/tex]
So, from the term [tex]$7x^2$[/tex] we obtain:
[tex]$$
63x^3 - 28x^2.
$$[/tex]
2. Multiply the second term, [tex]$9x$[/tex], by each term in the second polynomial:
[tex]$$
9x \cdot 9x = 81x^2 \quad \text{and} \quad 9x \cdot (-4) = -36x.
$$[/tex]
This gives:
[tex]$$
81x^2 - 36x.
$$[/tex]
3. Multiply the third term, [tex]$7$[/tex], by each term in the second polynomial:
[tex]$$
7 \cdot 9x = 63x \quad \text{and} \quad 7 \cdot (-4) = -28.
$$[/tex]
From this, we get:
[tex]$$
63x - 28.
$$[/tex]
4. Now, combine all the resulting terms:
[tex]$$
63x^3 - 28x^2 + 81x^2 - 36x + 63x - 28.
$$[/tex]
5. Combine like terms:
- The [tex]$x^3$[/tex] term is:
[tex]$$
63x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
-28x^2 + 81x^2 = 53x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
-36x + 63x = 27x.
$$[/tex]
- The constant is:
[tex]$$
-28.
$$[/tex]
Thus, the product of the polynomials is:
[tex]$$
63x^3 + 53x^2 + 27x - 28.
$$[/tex]
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