Problem 36 - Algebra and Trigonometry

Question

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ Q \text { has degree } 3, \text { and zeros }-3 \text { and } 1+i $$

Step-by-Step Solution

Answer

The polynomial is \(Q(x) = x^3 + x^2 - 4x + 6\).

Step 1: Understand the Roots

The polynomial is given to have three roots: \(-3\), \(1+i\), and the conjugate of \(1+i\), which is \(1-i\). Complex roots of polynomials with real coefficients occur in conjugate pairs.

Step 2: Construct Polynomial from Roots

Using the roots \(-3\), \(1+i\), and \(1-i\), the factors of the polynomial can be constructed. The polynomial can be written as \(Q(x) = (x + 3)((x - (1 + i))(x - (1 - i)))\).

Step 3: Simplify the Complex Factors

Simplify the factors involving the complex roots: \[(x - (1 + i))(x - (1 - i)) = ((x-1) - i)((x-1) + i)\]This simplifies to \[(x-1)^2 - i^2 = (x-1)^2 + 1 = (x^2 - 2x + 1 + 1) = x^2 - 2x + 2\].

Step 4: Form the Polynomial Expression

Now, the polynomial is \(Q(x) = (x + 3)(x^2 - 2x + 2)\).

Step 5: Expand the Polynomial

Expand the polynomial by distributing:\[Q(x) = x(x^2 - 2x + 2) + 3(x^2 - 2x + 2)\].Simplifying it will give:\[= x^3 - 2x^2 + 2x + 3x^2 - 6x + 6\].

Step 6: Combine Like Terms

Combine like terms to form the final expanded polynomial:\[Q(x) = x^3 + x^2 - 4x + 6\].

Key Concepts

Integer Coefficients

When we talk about integer coefficients in a polynomial, we mean that all the numbers multiplying the variables have to be whole numbers, like -2, 0, 7, and so on. This is crucial because integer coefficients often indicate that the polynomial is nicely behaved over the set of numbers known as integers. For algebra students, working with integer coefficients helps ensure that operations like addition and multiplication, which follow predictable rules, won't yield unexpected results.

• An example of a polynomial with integer coefficients is: \( x^3 + x^2 - 4x + 6 \).
• Each term in this polynomial: \( x^3 \), \( x^2 \), \( -4x \), and \( 6 \) is multiplied by integers.

Using integer coefficients is extremely useful when you aim to factor polynomials or when looking for rational roots using methods such as the Rational Root Theorem.

Complex Roots

In mathematics, a complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined by the property that \(i^2 = -1\). When a polynomial has complex roots, it means that some solutions for the polynomial equation are complex numbers. These roots are particularly interesting because they often occur when solving quadratic or higher-degree polynomial equations with real coefficients.

• In our exercise, \(1+i\) is a complex root of the polynomial.
• Complex roots in quadratic or cubic functions often appear when the discriminant (the part under the square root in the quadratic formula: \(b^2 - 4ac\)) is negative.

These roots broaden the range of potential solutions, allowing for any polynomial to be factorable over the complex numbers.

Conjugate Pairs

When dealing with polynomials that have real coefficients, if one complex root is a non-real number, such as \(1 + i\), then its complex conjugate, \(1 - i\), must also be a root. This leads to what are known as conjugate pairs. They are pairs of numbers of the form \(a + bi\) and \(a - bi\). Conjugate pairs are essential because they maintain the polynomial with real coefficients, a crucial requirement in many mathematical contexts.

• Having complex roots in conjugate pairs ensures the polynomial's coefficients remain real and thus more practical for real-world applications.
• A key property is that the product of a complex number and its conjugate is a real number: \((1+i)(1-i)=1^2-i^2=2\).

By including both parts of the pair as roots, we can construct polynomials that are guaranteed to have integer coefficients, adhering to the mathematical requirements for the problem at hand.