Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

A polynomial function \(f\) with real coefficients has the given degree, zeros, and solution point. Write the function (a) in completely factored form and (b) in polynomial form. Degree 3 Zeros $$-2,2+2 \sqrt{2} i$$ Solution Point $$f(-1)=-34$$

Step-by-Step Solution

Answer

The polynomial function in factored form is \(f(x) = -34(-4)(x+2)\) and in polynomial form is \(f(x) = 136x + 272\).

Step 1: Write the Factored Form

The factored form of a polynomial can be written using the zeros of the function. If a polynomial function has zeros \(p,q,r\), then the function can be written in factored form as: \(f(x) = a(x - p)(x - q)(x - r)\), where a is a constant.\n Now, as given the zeros of the function are -2, \(2+2\sqrt{2}i\) and \(2-2\sqrt{2}i\), we can substitute these zeros into the function, and the factored form of the function becomes \(f(x) = a(x+2)(x-(2+2\sqrt{2}i))(x-(2-2\sqrt{2}i))\).

Step 2: Simplify the Complex Terms

The factored form of the function contains complex numbers. To simplify this expression, we can use the fact that the product of a complex number and its conjugate is a real number. That is, \((2+2\sqrt{2}i))(2-2\sqrt{2}i)) = 4 - 8 = -4\). To simplify further, use the result to rewrite the factored form becomes \(f(x) = a(x+2)(-4)\), that is \(f(x) = -4a(x+2)\)

Step 3: Find the Coefficient

Now, we can use the given solution point, which is f(-1)=-34. Substitute x=-1 into the equation to find the value of constant a. \(-34 = -4a(-1+2)\). This results in a=-34 which leads to \(f(x) = -34(-4)(x+2)\).

Step 4: Write in Polynomial Form

To write the function in polynomial form, expand the simplified function \(f(x) = -34(-4)(x+2)\). Distributing gives \(f(x) = 136x + 272\).

Key Concepts

Factored Form of Polynomials

Understanding the factored form of a polynomial is crucial for solving various algebraic problems, including finding the polynomial's zeros. The factored form expresses a polynomial as a product of its linear factors based on its zeros.

For example, if a polynomial has zeros at values 'p', 'q', and 'r', then the polynomial can be represented in factored form as: \( f(x) = a(x - p)(x - q)(x - r) \), where 'a' is a non-zero constant known as the leading coefficient. It captures the polynomial's vertical stretch or shrink, as well as its orientation (inverted when 'a' is negative).

The zeros provided can be real numbers, as well as complex numbers which need special treatment and introduce the concept of complex conjugate pairs. It's important to recognize when complex numbers are zeros, their conjugate must also be a zero.

In the given exercise, the polynomial has a real zero at -2 and a pair of complex zeros. To express this in factored form, one starts by setting up the factors using the zeros provided: \( f(x) = a(x + 2)(x - (2 + 2\sqrt{2}i))(x - (2 - 2\sqrt{2}i)) \).

Complex Conjugate Pairs

Within polynomial equations, complex zeros often occur in conjugate pairs, especially when the polynomial has real coefficients. This property is vital for simplifying the complex factors in a polynomial's factored form. A complex conjugate pair consists of two complex numbers of the form \( a + bi \) and \( a - bi \), where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit.

When multiplying these conjugate pairs, the resulting product is always a real number. This is due to the fact that the product of 'i' and '-i' is -1, which eliminates the imaginary parts: \( (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \).

In the solution provided, the zeros \( 2 + 2\sqrt{2}i \) and \( 2 - 2\sqrt{2}i \) are conjugate pairs, and when multiplied together, they simplify to a real number, which is -4 in this particular case. This step significantly simplifies the polynomial to involve only real numbers, facilitating further operations such as expansion or finding the leading coefficient 'a'.

Polynomial Function Degree

The degree of a polynomial function is an essential characteristic that informs about its behavior and shape. The degree is the highest exponent of the variable in the polynomial when it is expressed in its standard form. For instance, a polynomial of degree three, often called a cubic polynomial, can be written in general as \( ax^3 + bx^2 + cx + d \).

The degree of the polynomial determines the maximum number of zeros it can have, the general shape of its graph, the number of turning points, and the end behavior of the function as x approaches infinity or negative infinity.

In the given exercise, the polynomial function is of degree three; therefore, it is expected to have up to three zeros and up to two turning points on its graph. The problem provides us with these zeros: -2, \( 2 + 2\sqrt{2}i \), and its complex conjugate. From these zeros and the degree, we can construct the appropriate factored form and consequently, determine the polynomial in its expanded standard form.