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Answer :
To multiply the given rational expressions and provide the answer in factored form, we need to factor each part of the expression and simplify. Let's go through it step-by-step:
Given expression:
[tex]\[
\frac{c^2 + 14c + 49}{10c^2 + 17c + 3} \cdot \frac{20c^2 + 9c + 1}{c^2 + 4c - 21}
\][/tex]
Step 1: Factor each polynomial.
1. Factor [tex]\(c^2 + 14c + 49\)[/tex]:
This is a perfect square trinomial:
[tex]\[
c^2 + 14c + 49 = (c + 7)(c + 7)
\][/tex]
2. Factor [tex]\(10c^2 + 17c + 3\)[/tex]:
Find factors of the form [tex]\((ac + b)(dc + e)\)[/tex] that match:
[tex]\[
10c^2 + 17c + 3 = (5c + 1)(2c + 3)
\][/tex]
3. Factor [tex]\(20c^2 + 9c + 1\)[/tex]:
This also factors as:
[tex]\[
20c^2 + 9c + 1 = (5c + 1)(4c + 1)
\][/tex]
4. Factor [tex]\(c^2 + 4c - 21\)[/tex]:
This factors as:
[tex]\[
c^2 + 4c - 21 = (c + 7)(c - 3)
\][/tex]
Step 2: Write the expression with factors.
Replace each polynomial with its factored form:
[tex]\[
\frac{(c + 7)(c + 7)}{(5c + 1)(2c + 3)} \cdot \frac{(5c + 1)(4c + 1)}{(c + 7)(c - 3)}
\][/tex]
Step 3: Cancel common factors.
- Cancel one [tex]\((c + 7)\)[/tex] from the numerator and the denominator.
- Cancel [tex]\((5c + 1)\)[/tex] from the numerator and the denominator.
After canceling, the expression becomes:
[tex]\[
\frac{(c + 7)(4c + 1)}{(2c + 3)(c - 3)}
\][/tex]
Thus, the answer in factored form is:
- Numerator: [tex]\((c + 7)(4c + 1)\)[/tex]
- Denominator: [tex]\((2c + 3)(c - 3)\)[/tex]
This is the simplified version of the multiplied rational expressions in factored form.
Given expression:
[tex]\[
\frac{c^2 + 14c + 49}{10c^2 + 17c + 3} \cdot \frac{20c^2 + 9c + 1}{c^2 + 4c - 21}
\][/tex]
Step 1: Factor each polynomial.
1. Factor [tex]\(c^2 + 14c + 49\)[/tex]:
This is a perfect square trinomial:
[tex]\[
c^2 + 14c + 49 = (c + 7)(c + 7)
\][/tex]
2. Factor [tex]\(10c^2 + 17c + 3\)[/tex]:
Find factors of the form [tex]\((ac + b)(dc + e)\)[/tex] that match:
[tex]\[
10c^2 + 17c + 3 = (5c + 1)(2c + 3)
\][/tex]
3. Factor [tex]\(20c^2 + 9c + 1\)[/tex]:
This also factors as:
[tex]\[
20c^2 + 9c + 1 = (5c + 1)(4c + 1)
\][/tex]
4. Factor [tex]\(c^2 + 4c - 21\)[/tex]:
This factors as:
[tex]\[
c^2 + 4c - 21 = (c + 7)(c - 3)
\][/tex]
Step 2: Write the expression with factors.
Replace each polynomial with its factored form:
[tex]\[
\frac{(c + 7)(c + 7)}{(5c + 1)(2c + 3)} \cdot \frac{(5c + 1)(4c + 1)}{(c + 7)(c - 3)}
\][/tex]
Step 3: Cancel common factors.
- Cancel one [tex]\((c + 7)\)[/tex] from the numerator and the denominator.
- Cancel [tex]\((5c + 1)\)[/tex] from the numerator and the denominator.
After canceling, the expression becomes:
[tex]\[
\frac{(c + 7)(4c + 1)}{(2c + 3)(c - 3)}
\][/tex]
Thus, the answer in factored form is:
- Numerator: [tex]\((c + 7)(4c + 1)\)[/tex]
- Denominator: [tex]\((2c + 3)(c - 3)\)[/tex]
This is the simplified version of the multiplied rational expressions in factored form.
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