Problem 39 - Algebra and Trigonometry

Question

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ T \text { has degree } 4, \text { zeros } i \text { and } 1+i, \text { and constant term } 12 $$

Step-by-Step Solution

Answer

The polynomial is \(6x^4 - 12x^3 + 18x^2 - 12x + 12\).

Step 1: Establish the zeros of the polynomial

Given zeros are \(i\) and \(1+i\). Since the coefficients must be integers, the polynomial must also include the conjugates of any complex zeros. Therefore, the zeros of the polynomial are \(i\), \(-i\), \(1+i\), and \(1-i\).

Step 2: Create factors from the zeros

Each zero gives rise to a factor. The factors are \((x - i)\), \((x + i)\), \((x - (1+i))\), and \((x - (1-i))\).

Step 3: Multiply factors with conjugate pairs

First, let's multiply the conjugate pairs to form real-coefficient quadratics:\[(x - i)(x + i) = x^2 + 1\]\[(x - (1+i))(x - (1-i)) = ((x-1)-i)((x-1)+i) = (x-1)^2 + 1 = x^2 - 2x + 2\]

Step 4: Multiply the resulting quadratics

Multiply the two quadratics obtained in Step 3:\[ (x^2 + 1)(x^2 - 2x + 2) \]

Step 5: Expand the polynomial

Expanding the expression:\[(x^2 + 1)(x^2 - 2x + 2) = x^4 - 2x^3 + 2x^2 + x^2 - 2x + 2 = x^4 - 2x^3 + 3x^2 - 2x + 2\]

Step 6: Adjust to satisfy the constant term

The current polynomial is \(x^4 - 2x^3 + 3x^2 - 2x + 2\). To make the constant term 12, multiply the entire polynomial by 6:\[6(x^4 - 2x^3 + 3x^2 - 2x + 2) = 6x^4 - 12x^3 + 18x^2 - 12x + 12\]

Step 7: Ensure polynomial satisfies all conditions

Check the polynomial:- Degree: 4- Zeros: \(i, -i, 1+i, 1-i\)- Constant Term: 12The polynomial \(6x^4 - 12x^3 + 18x^2 - 12x + 12\) meets all conditions, so it is the correct answer.

Key Concepts

Complex Zeros

In polynomial equations, a zero or root is the value that makes the whole polynomial equal to zero. A complex zero, such as \(i\) or \(1+i\), is not a real number. The imaginary unit \(i\) is defined as \(\sqrt{-1}\). When faced with a polynomial equation that includes such zeros, additional steps are needed to ensure integrity with integer coefficients.
For example, the presence of \(i\) or any other complex number in the zero set of a polynomial indicates that its conjugate must also be present. Complex numbers come in pairs like \(i\) and its conjugate \(-i\), which, when multiplied together, result in a real number. This step is essential if the polynomial is to have integer coefficients. Remember, only complex numbers in the form \(a + bi\) where \(b eq 0\) can be complex zeros.

Conjugate Pairs

When dealing with polynomials containing complex zeros, conjugate pairs play a significant role. These are pairs of complex numbers that consist of the same real part and opposite imaginary parts.
So if \(1+i\) is a zero, its conjugate zero \(1-i\) must also be included, as shown in the solution steps. This ensures that the resulting polynomial will have real coefficients.

• For a complex zero with the form \(a+bi\), its conjugate is \(a-bi\).
• The multiplication of conjugates always eliminates the imaginary part, resulting in a quadratic polynomial. For instance, \((x - (1+i))(x - (1-i)) = x^2 - 2x + 2\).

Conjugate pairs help maintain real values in polynomials, which is crucial for polynomials with integer coefficients.

Quadratic Polynomials

A quadratic polynomial is one of the simplest forms of polynomials, typically represented as \(ax^2 + bx + c\). When the factors derived from complex zeros are multiplied as conjugate pairs, the result is a quadratic polynomial with real coefficients.

• For example, the zeros \(i\) and \(-i\) give the quadratic factor \((x - i)(x + i) = x^2 + 1\).
• Similarly, \((x - (1+i))(x - (1-i))\) simplifies to the quadratic \(x^2 - 2x + 2\).

In constructing a higher-degree polynomial with integer coefficients, aggregating quadratic polynomials derived from complex roots is both efficient and necessary. This allows for easy multiplication and expansion in further steps of building the final polynomial.

Polynomial Expansion

Polynomial expansion involves multiplying polynomial expressions to obtain a single expanded polynomial. In the given exercise, the factors are first reduced by calculating products of conjugate pairs, resulting in simple quadratics, then expanded further.
When these quadratics \((x^2 + 1)\) and \((x^2 - 2x + 2)\) are multiplied, the expanded form becomes \(x^4 - 2x^3 + 3x^2 - 2x + 2\). This process involves distributing each term in the first quadratic polynomial with all terms in the second polynomial, combining like terms for simplification.

• Ensure that each multiplication adheres to the distributive property, which states: \(a(b+c) = ab + ac\).
• Combining the results helps achieve the resultant polynomial arranged in descending order of power.

Finally, this expanded polynomial is adjusted by a scalar to satisfy specific conditions, like matching a constant term, ensuring all the initial requirements are satisfied.