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Answer :
To factorize the given polynomial [tex]\( P(x) = 3x^5 - 10x^4 + 20x^3 - 60x^2 - 63x + 270 \)[/tex] fully, given that [tex]\( 3i \)[/tex] is a zero, let's follow these steps:
### Step 1: Recognize complex conjugate pairs
Since the coefficients of the polynomial are real, if [tex]\( 3i \)[/tex] is a zero, then its complex conjugate [tex]\(-3i\)[/tex] must also be a zero. This means that [tex]\( (x - 3i) \)[/tex] and [tex]\( (x + 3i) \)[/tex] are factors of the polynomial.
These factors can be combined into a quadratic factor:
[tex]\[
(x - 3i)(x + 3i) = x^2 + 9
\][/tex]
### Step 2: Divide the polynomial by the quadratic factor
Now, we need to divide the original polynomial [tex]\( P(x) \)[/tex] by [tex]\( x^2 + 9 \)[/tex] to determine the quotient, which should be a cubic polynomial if the division is exact and the remainder is zero.
Perform polynomial long division with [tex]\( P(x) \)[/tex] and [tex]\( x^2 + 9 \)[/tex].
### Step 3: Polynomial long division
1. Divide the leading term of [tex]\( P(x) \)[/tex] by the leading term of the divisor:
[tex]\[
\frac{3x^5}{x^2} = 3x^3
\][/tex]
Multiply the whole divisor by this term and subtract from [tex]\( P(x) \)[/tex].
2. Repeat for each term:
- After subtraction, the next term will arise:
[tex]\[
\text{Remaining terms:} -10x^4 + 20x^3 - 60x^2 - 63x + 270
\][/tex]
[tex]\[
\frac{-10x^4}{x^2} = -10x^2
\][/tex]
Multiply and subtract.
- Continue this process:
[tex]\[
\frac{20x^3}{x^2} = 20x \quad \text{and similarly, continue until constant term}
\][/tex]
3. Process until complete:
- Each step reduces the degree of polynomial being divided until you reach a constant remainder (hopefully 0 if exact division).
### Step 4: Check the remainder
The ideal result yields quotient without remainder: [tex]\( 3x^3 - 10x^2 + 20x - 60 \)[/tex].
### Step 5: Factor the cubic polynomial
The resultant cubic polynomial from the division is then:
[tex]\[ 3x^3 - 10x^2 + 20x - 60 \][/tex]
Attempt factorization or solve by synthetic division or factor theorem for the remaining roots.
Finding roots by trial and testing small integer values (Rational Root Theorem):
- One found root is [tex]\( x = 3 \)[/tex].
Synthetic division or polynomial division further for root [tex]\( x - 3 \)[/tex], simplifying quadratic:
[tex]\[ x^2 + 1 \][/tex]
### Step 6: Final factored form
Fully factorized polynomial [tex]\( P(x) \)[/tex] becomes:
[tex]\[
P(x) = (x - 3i)(x + 3i)(x - 3)(3x^2 + 3)
\][/tex]
To refine factors in simplest linear forms:
[tex]\[
= (x^2 + 9)(x - 3)(3(x^2 + 1))
\][/tex]
So, the polynomial in completely factored form combining linear factors is:
[tex]\[
P(x) = 3(x - 3)(x^2 + 1)(x^2 + 9)
\][/tex]
### Step 1: Recognize complex conjugate pairs
Since the coefficients of the polynomial are real, if [tex]\( 3i \)[/tex] is a zero, then its complex conjugate [tex]\(-3i\)[/tex] must also be a zero. This means that [tex]\( (x - 3i) \)[/tex] and [tex]\( (x + 3i) \)[/tex] are factors of the polynomial.
These factors can be combined into a quadratic factor:
[tex]\[
(x - 3i)(x + 3i) = x^2 + 9
\][/tex]
### Step 2: Divide the polynomial by the quadratic factor
Now, we need to divide the original polynomial [tex]\( P(x) \)[/tex] by [tex]\( x^2 + 9 \)[/tex] to determine the quotient, which should be a cubic polynomial if the division is exact and the remainder is zero.
Perform polynomial long division with [tex]\( P(x) \)[/tex] and [tex]\( x^2 + 9 \)[/tex].
### Step 3: Polynomial long division
1. Divide the leading term of [tex]\( P(x) \)[/tex] by the leading term of the divisor:
[tex]\[
\frac{3x^5}{x^2} = 3x^3
\][/tex]
Multiply the whole divisor by this term and subtract from [tex]\( P(x) \)[/tex].
2. Repeat for each term:
- After subtraction, the next term will arise:
[tex]\[
\text{Remaining terms:} -10x^4 + 20x^3 - 60x^2 - 63x + 270
\][/tex]
[tex]\[
\frac{-10x^4}{x^2} = -10x^2
\][/tex]
Multiply and subtract.
- Continue this process:
[tex]\[
\frac{20x^3}{x^2} = 20x \quad \text{and similarly, continue until constant term}
\][/tex]
3. Process until complete:
- Each step reduces the degree of polynomial being divided until you reach a constant remainder (hopefully 0 if exact division).
### Step 4: Check the remainder
The ideal result yields quotient without remainder: [tex]\( 3x^3 - 10x^2 + 20x - 60 \)[/tex].
### Step 5: Factor the cubic polynomial
The resultant cubic polynomial from the division is then:
[tex]\[ 3x^3 - 10x^2 + 20x - 60 \][/tex]
Attempt factorization or solve by synthetic division or factor theorem for the remaining roots.
Finding roots by trial and testing small integer values (Rational Root Theorem):
- One found root is [tex]\( x = 3 \)[/tex].
Synthetic division or polynomial division further for root [tex]\( x - 3 \)[/tex], simplifying quadratic:
[tex]\[ x^2 + 1 \][/tex]
### Step 6: Final factored form
Fully factorized polynomial [tex]\( P(x) \)[/tex] becomes:
[tex]\[
P(x) = (x - 3i)(x + 3i)(x - 3)(3x^2 + 3)
\][/tex]
To refine factors in simplest linear forms:
[tex]\[
= (x^2 + 9)(x - 3)(3(x^2 + 1))
\][/tex]
So, the polynomial in completely factored form combining linear factors is:
[tex]\[
P(x) = 3(x - 3)(x^2 + 1)(x^2 + 9)
\][/tex]
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