Problem 35 - Precalculus : Building Concepts and Connections

Question

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$x^{4}-5 x^{3}+7 x^{2}-5 x+6 ; \text { zero: } x=2$$

Step-by-Step Solution

Answer

The resulting expression should be a product of four linear factors, each of which corresponds to a root of the polynomial. It might contain complex coefficients if there are complex roots, but it's also possible that all roots are real.

Step 1: Use synthetic division to simplify the polynomial

Begin by taking the given root \(x = 2\). Now, apply synthetic division to divide the polynomial \(x^{4} - 5x^{3} + 7x^{2} - 5x + 6\) by the linear factor \(x - 2\). Write down the coefficients of the polynomial above the synthetic division, and then perform the synthetic division process. It will give a new polynomial of degree 3.

Step 2: Find the roots of the cubic polynomial

A cubic polynomial can have one or three roots. The cubic polynomial in this case should be expressed in terms of \(x\), and then you need to find the roots. The roots can be real or complex. In terms of real roots, the cubic polynomial might be factorized into one linear term and one quadratic term, or might be factorized into three linear terms. If it cannot be factored into real roots, use the quadratic formula or numerical methods to find the complex roots.

Step 3: Express the original polynomial as product of linear factors

Now that all four roots are known, the original quartic polynomial can be expressed as a product of four linear factors. A root \(r\) of a polynomial corresponds to a factor of \((x - r)\). To get the polynomial back in the form given, the coefficients of the linear factors might be complex numbers.

Key Concepts

Complex Numbers

Complex numbers play a significant role in polynomial factorization, especially when dealing with non-real roots. A complex number is of the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part, with \( i \) being the imaginary unit defined as \( i^2 = -1 \).

When factoring polynomials such as the one given in the exercise, it is possible for some or all of the roots to be complex. This occurs when the discriminant in the quadratic formula is negative, resulting in the square root of a negative value. For instance, if you have a quadratic equation from the polynomial's factorization, and you apply the quadratic formula, the roots may be complex depending on the calculator under the square root sign.

Using complex numbers allows us to express any polynomial as a product of linear factors, extending the realm of solutions beyond real numbers. This makes complex numbers essential for completely solving polynomials like \( x^4 - 5x^3 + 7x^2 - 5x + 6 \). Understanding complex roots helps in breaking down polynomials into simpler, more manageable pieces.

Synthetic Division

Synthetic division is a simplified form of polynomial division, mainly used when dividing by a linear factor of the form \( x - r \). It is a streamlined process that reduces the potential for arithmetic errors, making it a popular choice among students and mathematicians alike.

To perform synthetic division, you only need the coefficients of the polynomial and the root or zero given as \( r \) in \( x - r \). Write the coefficients on a line, then write the root, \( r \), to the left. This process includes multiplying, adding, and bringing down results, until all terms have been processed.

• Write coefficients of the polynomial.
• Use the given zero, in this case, \( x = 2 \).
• Follow the multiplication and addition process of synthetic division.

After you complete synthetic division, you obtain the coefficients for a lower degree polynomial, typically one degree less. This is crucial in polynomial root finding and further factorization, as it allows us to simplify and analyze the polynomial's remaining parts.

Polynomial Roots

Finding polynomial roots is a central aspect of factorizing polynomials into linear factors, and it involves determining the values of \( x \) where the polynomial evaluates to zero. Understanding this concept allows us to express a polynomial as products of terms like \( (x - r) \), where \( r \) is a root.

For the given polynomial \( x^4 - 5x^3 + 7x^2 - 5x + 6 \), starting with a known root such as \( x = 2 \) drastically simplifies this process. Once we use synthetic division to remove the given factor, we are left with a cubic polynomial.

• Consider possible real and complex roots.
• Employ methods like the Rational Root Theorem or the quadratic formula when applicable.
• Use numerical solutions or computer algebra systems for exact complex roots if necessary.

By identifying all roots, including both real and complex, the polynomial can be completely decomposed into linear factors. This complete decomposition gives comprehensive insight into the behavior of the polynomial function.