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Answer :
To factor the polynomial [tex]\( c^4 + 6x^3 + 33x^2 + 150x + 200 \)[/tex] using synthetic division, let's go through a step-by-step process.
### Step 1: Identify Possible Rational Roots
Using the Rational Root Theorem, identify potential rational roots by taking the factors of the constant term (200) and dividing them by the factors of the leading coefficient (1). This gives us possible roots such as ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100, ±200.
### Step 2: Test Possible Roots Using Synthetic Division
We need to test these roots incrementally using synthetic division until we find one that results in a zero remainder, indicating it is a root of the polynomial.
Testing -4:
Perform synthetic division with [tex]\( -4 \)[/tex]. The resulting remainder is zero, so [tex]\( x + 4 \)[/tex] is a factor.
### Step 3: Factor the Polynomial
Since we found that [tex]\( x + 4 \)[/tex] is a factor, divide the polynomial by [tex]\( x + 4 \)[/tex]:
1. Using synthetic division, divide [tex]\( c^4 + 6x^3 + 33x^2 + 150x + 200 \)[/tex] by [tex]\( x + 4 \)[/tex]. The remaining polynomial after division is [tex]\( x^3 + 2x^2 + 25x + 50 \)[/tex].
### Step 4: Repeat for the Remaining Polynomial
Now, apply synthetic division again:
Testing -2:
With synthetic division of the cubic [tex]\( x^3 + 2x^2 + 25x + 50 \)[/tex] by [tex]\( x + 2 \)[/tex], you find that it is also a factor.
### Step 5: Complete the Factorization
Now you have further divided the polynomial to:
- The initial division gave remainder zero and the quotient [tex]\( (x^3 + 2x^2 + 25x + 50) \)[/tex].
- The next division gave remainder zero and the quotient [tex]\( (x^2 + 25) \)[/tex].
Therefore, the fully factored form of the polynomial is:
[tex]\[ (x + 2)(x + 4)(x^2 + 25) \][/tex]
### Conclusion
The correct option for the factored form of the polynomial [tex]\( c^4 + 6x^3 + 33x^2 + 150x + 200 \)[/tex] is:
C. [tex]\((x + 2)(x + 4)(x^2 + 25)\)[/tex]
### Step 1: Identify Possible Rational Roots
Using the Rational Root Theorem, identify potential rational roots by taking the factors of the constant term (200) and dividing them by the factors of the leading coefficient (1). This gives us possible roots such as ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100, ±200.
### Step 2: Test Possible Roots Using Synthetic Division
We need to test these roots incrementally using synthetic division until we find one that results in a zero remainder, indicating it is a root of the polynomial.
Testing -4:
Perform synthetic division with [tex]\( -4 \)[/tex]. The resulting remainder is zero, so [tex]\( x + 4 \)[/tex] is a factor.
### Step 3: Factor the Polynomial
Since we found that [tex]\( x + 4 \)[/tex] is a factor, divide the polynomial by [tex]\( x + 4 \)[/tex]:
1. Using synthetic division, divide [tex]\( c^4 + 6x^3 + 33x^2 + 150x + 200 \)[/tex] by [tex]\( x + 4 \)[/tex]. The remaining polynomial after division is [tex]\( x^3 + 2x^2 + 25x + 50 \)[/tex].
### Step 4: Repeat for the Remaining Polynomial
Now, apply synthetic division again:
Testing -2:
With synthetic division of the cubic [tex]\( x^3 + 2x^2 + 25x + 50 \)[/tex] by [tex]\( x + 2 \)[/tex], you find that it is also a factor.
### Step 5: Complete the Factorization
Now you have further divided the polynomial to:
- The initial division gave remainder zero and the quotient [tex]\( (x^3 + 2x^2 + 25x + 50) \)[/tex].
- The next division gave remainder zero and the quotient [tex]\( (x^2 + 25) \)[/tex].
Therefore, the fully factored form of the polynomial is:
[tex]\[ (x + 2)(x + 4)(x^2 + 25) \][/tex]
### Conclusion
The correct option for the factored form of the polynomial [tex]\( c^4 + 6x^3 + 33x^2 + 150x + 200 \)[/tex] is:
C. [tex]\((x + 2)(x + 4)(x^2 + 25)\)[/tex]
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