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Answer :
To find the remaining zeros of the polynomial [tex]\( h(x) = 2x^4 + 3x^3 + 70x^2 + 108x - 72 \)[/tex], given that one zero is [tex]\(-6i\)[/tex], let's follow these steps:
1. Understand complex conjugate pairs:
- Since polynomials with real coefficients have zeros that occur in conjugate pairs when they are complex, this means that since [tex]\(-6i\)[/tex] is a zero, its conjugate [tex]\(6i\)[/tex] is also a zero.
2. Identify the possible roots:
- Given the degree of the polynomial is 4, and we already have two zeros, [tex]\(-6i\)[/tex] and [tex]\(6i\)[/tex], the polynomial must have two more zeros.
- These zeros will be real numbers because complex zeros come in pairs in real polynomials.
3. Find the remaining zeros:
- Let's determine the polynomial that results when you factor out [tex]\( (x + 6i)(x - 6i) \)[/tex] from the original polynomial. The factors from the complex zeros are multiplied to form a quadratic factor: [tex]\( (x + 6i)(x - 6i) = x^2 + 36 \)[/tex].
- Divide the original polynomial by this quadratic to find the other factor, which is a real quadratic polynomial.
- Solve the real quadratic factor to find the remaining zeros, which are real numbers.
4. Determine the remaining real zeros:
- After factoring and solving the quadratic from the division of [tex]\( h(x) \)[/tex] by [tex]\( x^2 + 36 \)[/tex], you find the remaining zeros, which are [tex]\(-2\)[/tex] and [tex]\(0.5\)[/tex].
So, the remaining zeros of the polynomial [tex]\( h(x) \)[/tex] are [tex]\(-2\)[/tex] and [tex]\(0.5\)[/tex].
1. Understand complex conjugate pairs:
- Since polynomials with real coefficients have zeros that occur in conjugate pairs when they are complex, this means that since [tex]\(-6i\)[/tex] is a zero, its conjugate [tex]\(6i\)[/tex] is also a zero.
2. Identify the possible roots:
- Given the degree of the polynomial is 4, and we already have two zeros, [tex]\(-6i\)[/tex] and [tex]\(6i\)[/tex], the polynomial must have two more zeros.
- These zeros will be real numbers because complex zeros come in pairs in real polynomials.
3. Find the remaining zeros:
- Let's determine the polynomial that results when you factor out [tex]\( (x + 6i)(x - 6i) \)[/tex] from the original polynomial. The factors from the complex zeros are multiplied to form a quadratic factor: [tex]\( (x + 6i)(x - 6i) = x^2 + 36 \)[/tex].
- Divide the original polynomial by this quadratic to find the other factor, which is a real quadratic polynomial.
- Solve the real quadratic factor to find the remaining zeros, which are real numbers.
4. Determine the remaining real zeros:
- After factoring and solving the quadratic from the division of [tex]\( h(x) \)[/tex] by [tex]\( x^2 + 36 \)[/tex], you find the remaining zeros, which are [tex]\(-2\)[/tex] and [tex]\(0.5\)[/tex].
So, the remaining zeros of the polynomial [tex]\( h(x) \)[/tex] are [tex]\(-2\)[/tex] and [tex]\(0.5\)[/tex].
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