Problem 20
Question
Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}+4$$
Step-by-Step Solution
Verified Answer
The zeros of the polynomial \(x^{2}+4\) are \(2i\) and \(-2i\). The polynomial in factored form is \(p(x) = (x - 2i)(x + 2i)\).
1Step 1: Identifying the Zeros of the Polynomial
For finding the zeros of a polynomial, we set the polynomial to zero and solve for \(x\). So, we set: \(x^{2}+4=0\). Subtracting 4 from both sides, we get, \(x^{2}=-4\). Taking the square root of both sides yields \(x = \pm 2i\). Hence, the zeros of the polynomial are \(2i\) and \(-2i\), both of which are non-real numbers.
2Step 2: Expressing as Linear Factors
To express this polynomial as a product of linear factors, we rewrite each zero as a factor of the polynomial. Given the zeros \(2i\) and \(-2i\), the linear factors of \(p(x)\) are \(x - 2i\) and \(x + 2i\). Thus, \(p(x)\) can be written in factored form as \(p(x) = (x - 2i)(x + 2i)\).
Key Concepts
Complex ZerosLinear FactorsPolynomial EquationsImaginary Numbers
Complex Zeros
Understanding complex zeros is crucial when dealing with polynomials that don't cross the x-axis on a graph.
When we encounter an equation like \(x^2 + 4 = 0\), we find that its solutions aren't real numbers because squaring any real number won't give us a negative result. This is where complex zeros come into play. Complex zeros are solutions that include imaginary numbers, which are multiples of the imaginary unit, \(i\), defined as \(\sqrt{-1}\).
In our example, we found the zeros to be \(\pm 2i\), with \(i\) representing the imaginary unit. Complex zeros often come in pairs that are conjugates of each other, meaning they have the same real part but opposite imaginary parts, as seen in \(2i\) and \(−2i\).
It's important to know that polynomial equations with real coefficients will always have complex zeros in conjugate pairs, which assures us that when we find one complex zero, its conjugate is also a zero of the polynomial.
When we encounter an equation like \(x^2 + 4 = 0\), we find that its solutions aren't real numbers because squaring any real number won't give us a negative result. This is where complex zeros come into play. Complex zeros are solutions that include imaginary numbers, which are multiples of the imaginary unit, \(i\), defined as \(\sqrt{-1}\).
In our example, we found the zeros to be \(\pm 2i\), with \(i\) representing the imaginary unit. Complex zeros often come in pairs that are conjugates of each other, meaning they have the same real part but opposite imaginary parts, as seen in \(2i\) and \(−2i\).
It's important to know that polynomial equations with real coefficients will always have complex zeros in conjugate pairs, which assures us that when we find one complex zero, its conjugate is also a zero of the polynomial.
Linear Factors
Breaking down polynomials into linear factors is a method of expressing them as a product of their simplest components.
For any polynomial, a linear factor corresponds to each zero of the function. If \(p(x)\) is a polynomial with a zero at \(x = a\), then \(x - a\) is a linear factor of \(p(x)\). In the polynomial \(p(x) = x^2 + 4\), the zeros we found were \(2i\) and \(−2i\). Thus, the corresponding linear factors are \(x - 2i\) and \(x + 2i\).
Expressing the polynomial as \(p(x) = (x - 2i)(x + 2i)\) gives a clear picture of what values of \(x\) will make \(p(x)\) zero. It also demonstrates the Fundamental Theorem of Algebra, which states that a polynomial of degree \(n\) will have \(n\) linear factors.
For any polynomial, a linear factor corresponds to each zero of the function. If \(p(x)\) is a polynomial with a zero at \(x = a\), then \(x - a\) is a linear factor of \(p(x)\). In the polynomial \(p(x) = x^2 + 4\), the zeros we found were \(2i\) and \(−2i\). Thus, the corresponding linear factors are \(x - 2i\) and \(x + 2i\).
Expressing the polynomial as \(p(x) = (x - 2i)(x + 2i)\) gives a clear picture of what values of \(x\) will make \(p(x)\) zero. It also demonstrates the Fundamental Theorem of Algebra, which states that a polynomial of degree \(n\) will have \(n\) linear factors.
Polynomial Equations
When we speak of polynomial equations, we refer to mathematical expressions involving sums of powers of variables. These equations are fundamental in mathematics and can describe a variety of complex relationships.
The general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0\), where \(a_n \eq 0\) and all coefficients are real numbers. The greatest exponent \(n\) is the degree of the polynomial, which tells us the maximum number of zeros (solutions) it can have.
Solving polynomial equations like \(x^2+4=0\) involves finding all values of \(x\) that satisfy the equation, meaning when substituted into the polynomial, they result in a zero. These solutions can be either real or non-real (complex) and are essential for understanding the behavior of the polynomial function.
The general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0\), where \(a_n \eq 0\) and all coefficients are real numbers. The greatest exponent \(n\) is the degree of the polynomial, which tells us the maximum number of zeros (solutions) it can have.
Solving polynomial equations like \(x^2+4=0\) involves finding all values of \(x\) that satisfy the equation, meaning when substituted into the polynomial, they result in a zero. These solutions can be either real or non-real (complex) and are essential for understanding the behavior of the polynomial function.
Imaginary Numbers
At the heart of complex zeros lies the concept of imaginary numbers. Imaginary numbers expand our understanding of what numbers can be beyond just the real number line.
The building block of imaginary numbers is the imaginary unit \(i\), which is defined as \(\sqrt{-1}\). This is because in the set of real numbers, no number squared equals a negative. Therefore, \(i\) provides a way to extend the real numbers to include solutions to equations like \(x^2 = -1\).
From this unit, we get more complex numbers, such as \(2i\) or \(3i\), and so on. When we talk about calculating the zeros of polynomials, we sometimes find solutions that involve these imaginary numbers. Being comfortable with imaginary numbers is essential for students to grasp more advanced concepts in algebra, such as solving polynomial equations and transforming them into linear factors.
The building block of imaginary numbers is the imaginary unit \(i\), which is defined as \(\sqrt{-1}\). This is because in the set of real numbers, no number squared equals a negative. Therefore, \(i\) provides a way to extend the real numbers to include solutions to equations like \(x^2 = -1\).
From this unit, we get more complex numbers, such as \(2i\) or \(3i\), and so on. When we talk about calculating the zeros of polynomials, we sometimes find solutions that involve these imaginary numbers. Being comfortable with imaginary numbers is essential for students to grasp more advanced concepts in algebra, such as solving polynomial equations and transforming them into linear factors.
Other exercises in this chapter
Problem 19
Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$f(x)=-x^{3}+2 x$$
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Solve the polynomial inequality. $$x^{4}-3 x^{2}
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Find the domain and the vertical and horizontal asymptotes (if any). $$h(x)=\frac{x+2}{4+x^{2}}$$
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Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic divisi
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