91Ó°ÊÓ

Let’s start with understanding how to factorize a polynomial. Factorizing is the process of breaking down a polynomial into simpler components (called factors) that, when multiplied together, yield the original polynomial. Suppose we have a polynomial \( f(x) \). To factorize \( f(x) \), we look for its roots or zeros – the values of \( x \) that make \( f(x) = 0 \). Once we find the zeros, we can express \( f(x) \) as a product of linear factors and possibly other polynomial factors.

For instance, if one of the zeros is \( 3 - i \) and its complex conjugate \( 3 + i \), we can use these zeros to form factors like \( (x - (3 - i)) \) and \( (x - (3 + i)) \). When expanded and combined with other factors, they will yield the original polynomial.

In our exercise, we are given a complex number zero \( 3 - i \). Complex numbers have both a real part and an imaginary part. When you have polynomials with real coefficients, the complex roots always appear in conjugate pairs. This means if \( 3 - i \) is a zero, \( 3 + i \) is also a zero.

Complex conjugates are pairs of complex numbers of the form \( a + bi \) and \( a - bi \). They play a significant role in polynomial factorization because their product is always a quadratic polynomial with real coefficients, such as in our example:

• First zero: \(3 - i\)
• Conjugate zero: \(3 + i\)

Together, they form the quadratic factor \( (x - (3 - i))(x - (3 + i)) \), which simplifies to \( x^2 - 6x + 10 \).

Quadratic polynomials are in the form \( ax^2 + bx + c \). They have at most two roots, which can be real or complex. In solving polynomials like our example, we often end up simplifying the problem into a quadratic polynomial. Once we have a quadratic polynomial like \( x^2 - 6x + 10 \), we can solve it to find its roots using various methods:

• Factoring, if the polynomial can be written as a product of two binomials.
• The Quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square.

After forming a quadratic factor from the complex conjugate pair, the next step is to divide the original polynomial by this factor to reduce its degree. Polynomial division, much like numerical division, helps in peeling off the layers of polynomial complexities.

Let's perform the division for our polynomial \( f(x) = x^4 - 6x^3 + 5x^2 + 30x - 50 \) by \( x^2 - 6x + 10 \). This gives us a quotient that is easier to handle, usually reducing the polynomial’s degree by 2. Here's how it works:

• Determine how many times the divisor fits into the leading term of the dividend.
• Subtract, bring down the next term, and repeat until all terms are exhausted.

Finally, solving equations involves finding the zeros of a polynomial, which corresponds to the values of \( x \) where \( f(x) = 0 \). To solve the polynomial from our exercise, we:

• First find all the zeros, starting with given ones and using polynomial division to find others.
• Express the polynomial as a product of its linear factors.
• Set the factorized form equal to zero and solve for each \( x \).

For example, from our steps, we factorize the polynomial to get terms that look like \( (x - (3 - i)) \), \( (x - (3 + i)) \), and other possible factors. Solving these linear equations individually gives us all the zeros. This method ensures we cover all solutions efficiently.