Question

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22,-3+\sqrt{2} i$$

Short answer

The foots of the polynomial are \( -3 + \sqrt{2}i\), \( -3 - \sqrt{2}i\) and the solutions of the resulting quadratic polynomial from step 3.

Step-by-step solution

Use of Complex Conjugate Root Theorem

If \(-3 + \sqrt{2}i\) is a root of the polynomial, then according to the Complex Conjugate Root Theorem, the conjugate, \(-3 - \sqrt{2}i\), is also a root.

Perform polynomial division

Now, the given polynomial \(f(x) = x^{4} + 3x^{3} - 5x^{2} - 21x + 22\) is divided by the quadratic factor \((x - (-3+\sqrt{2}i))\) and \((x - (-3-\sqrt{2}i))\), which simplifies to \((x + 3 - \sqrt{2}i)\) and \((x + 3 + \sqrt{2}i)\). By doing this, the quotient obtained will be a quadratic polynomial.

Solve the quadratic polynomial

Set the resulting quadratic polynomial equal to zero and solve for \(x\). The solutions obtained will be the remaining roots of the polynomial \(f(x)\).

Key concepts

Polynomial Division

Polynomial division is akin to long division that we use with numbers, except that it involves dividing polynomials instead of integers. In the context of finding zeros of a polynomial, we often use division to simplify the polynomial by factoring out known zeros. For instance, given a polynomial like \(f(x) = x^{4} + 3x^{3} - 5x^{2} - 21x + 22\), and a zero such as \( -3+\sqrt{2}i\), we can create a factor corresponding to this zero and its complex conjugate.

Mathematically, to divide \(f(x)\) by the factors from zeros, we set up a division statement similar to long division. The process involves subtracting multiples of the divisor from the dividend to find the quotient and the remainder. When a zero of the polynomial is used to form the divisor, ideally, the remainder will be zero - showing that the divisor is indeed a factor of the original polynomial. This step significantly reduces the complexity of the polynomial, making it easier to find the remaining zeros.

Finding Zeros of a Polynomial

The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. These values are also referred to as roots or solutions.

Finding Complex Zeros

When we are given at least one complex zero, such as \( -3+\sqrt{2}i\), we can use the Complex Conjugate Root Theorem to determine another. The theorem states that if a polynomial has real coefficients, then every complex zero will have a conjugate pair.

After identifying the complex conjugate pairs, we can factor the polynomial into simpler parts. We often end up with a quadratic polynomial, which is much easier to solve. Solving the remaining quadratic polynomial gives us the additional zeros necessary to account for all the values of \(x\) that satisfy the equation \(f(x)=0\).

Quadratic Polynomial

When polynomial division is applied to a given polynomial using its complex zeros, we are frequently led to a quadratic polynomial. A quadratic polynomial has the form \(ax^2 + bx + c\), and it is the most basic polynomial that can have complex zeros.

Once we have a quadratic polynomial, finding its zeros becomes a more straightforward process, usually involving techniques such as factoring, completing the square, or applying the quadratic formula. In our context, solving the resulting quadratic polynomial after division allows us to identify the remaining zeros of the original higher-degree polynomial.

Complex Roots of Polynomials

Complex roots of polynomials occur in conjugate pairs if the polynomial has real coefficients, which is summarized in the Complex Conjugate Root Theorem. This fundamental principle means that if a polynomial has one complex root, such as \( -3+\sqrt{2}i\), its conjugate \( -3-\sqrt{2}i\) is guaranteed to be another root.

The theorem simplifies the process of finding all zeros since it automatically provides us with a pair of zeros upon discovering one complex zero. To find these complex roots, we can use various techniques, including polynomial division and solving quadratic polynomials, as demonstrated in the given exercise. By systematically applying these methods, we can locate all the roots of a polynomial and thus fully understand its behavior.