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Chapter 4: Problem 76

Use the given zero to completely factor \(P(x)\) into linear factors. Zero: \(2-i ; \quad P(x)=x^{4}-4 x^{3}+9 x^{2}-16 x+20\)

Short Answer

Expert verified
Factor: \((x - 2)^3 (x^2 - 4x + 5)\).

Step by step solution

01

Identify Complex Roots

Given the zero, which is a complex number in the form \(2-i\), its conjugate \(2+i\) is also a zero because polynomials with real coefficients have conjugate pairs as roots.
02

Form Quadratic Factor from Complex Roots

Using the complex zeros \(2-i\) and \(2+i\), we can create a quadratic factor: \[(x-(2-i))(x-(2+i)) = (x-2+i)(x-2-i) = (x-2)^2 - (i)^2 = (x-2)^2 + 1.\]This simplifies to \((x-2)^2 + 1 = x^2 - 4x + 5\).
03

Perform Polynomial Division

Divide the original polynomial \(P(x) = x^4 - 4x^3 + 9x^2 - 16x + 20\) by the quadratic factor \(x^2 - 4x + 5\) to find another factor. Use synthetic or long division to simplify the polynomial.The result of this division will be a quadratic polynomial factor.
04

Find Remaining Roots

The result of the division is \(x^2 - 4x + 4\).Notice this is a perfect square trinomial that can be factored further: \((x - 2)(x - 2) = (x - 2)^2\).
05

Combine All Factors

Combine all linear factors from the steps above:The complete factorization of \(P(x)\) is \[(x - (2-i))(x - (2+i))(x - 2)(x - 2),\]which simplifies to \((x - 2)^3 (x^2 -4x + 5)\), confirming all factors are linear with complex roots included.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by \(i^2 = -1\). These numbers are essential in algebra as they allow for the solutions of equations that have no real-number solutions.
They appear in many algebraic situations, especially when working with polynomials. For example, if you encounter a polynomial with real coefficients, finding zeros could lead you to complex numbers. In our exercise, the complex number provided is \(2-i\), which is one zero of the polynomial \(P(x)\).
Conjugate Roots
Conjugate roots are pairs of complex numbers that are interconnected. If a polynomial has a complex root \(a + bi\), its conjugate \(a - bi\) is also a root. This comes from the fact that polynomials with real coefficients must have real or conjugate pairs of complex roots.
Understanding conjugate roots is vital because they ensure all roots of a polynomial with real coefficients occur in pairs. In our example, the provided zero is \(2 - i\), which means its conjugate \(2 + i\) is also a zero. This is crucial in forming a quadratic factor as we'll see next.
Quadratic Factors
Quadratic factors are polynomial expressions of degree two, such as \(x^2 - 4x + 5\). They serve as building blocks for dividing higher-degree polynomials.
When a complex number and its conjugate are roots, they combine to create a quadratic factor with real coefficients. Using the zeros \(2-i\) and \(2+i\), we create the quadratic factor:
  • Find the product \((x - (2-i))(x - (2+i))\)
  • This simplifies to \((x - 2)^2 + 1 = x^2 - 4x + 5\)
This quadratic factor is essential for further dividing the polynomial to find other factors.
Polynomial Division
Polynomial division allows you to break down a large polynomial into smaller, more manageable factors. There are techniques like synthetic or long division to achieve this. In the exercise, given the polynomial \(P(x)\) is divided by the quadratic factor \(x^2 - 4x + 5\) to extract another factor.
By carefully performing the division:
  • The goal is simplifying \(P(x) = x^4 - 4x^3 + 9x^2 - 16x + 20\).
  • The division results in \(x^2 - 4x + 4\), a quadratic itself.
  • This further factors into \((x-2)^2\), revealing remaining linear factors.
Polynomial division aids in pinpointing remaining roots and fully factorizing the polynomial into linear terms.

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Most popular questions from this chapter

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=8 x^{3}+50 x^{2}+47 x-15 ; \quad k=-5$$

Answer true or false to each statement. Then support your answer by graphing. If a polynomial function of even degree has a negative leading coefficient and a positive \(y\) -value for its \(y\) -intercept, it must have at least two real zeros.

For each polynomial at least one zero is given. Find all others analytically. $$P(x)=x^{3}-2 x^{2}-5 x+6 ; 3$$

Use a graphing calculator to find a comprehensive graph and answer each of the following. (a) Determine the domain. (b) Determine all local minimum points, and tell if any is an absolute minimum point. (Approximate coordinates to the nearest hundredth.) (c) Determine all local maximum points, and tell if any is an absolute maximum point. (Approximate coordinates to the nearest hundredth.) (d) Determine the range. (If an approximation is necessary. give it to the nearest hundredth.) (e) Determine all intercepts. For each function, there is at least one \(x\) -intercept that has an integer x-value. For those that are not integers, give approximations to the nearest hundredth. Determine the \(y\) -intercept analytically. (f) Give the open interval(s) over which the function is increasing. (g) Give the open interval(s) over which the function is decreasing. $$P(x)=-3 x^{6}+2 x^{5}+9 x^{4}-8 x^{3}+11 x^{2}+4$$

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$\sqrt{6} ; \quad P(x)=-2 x^{6}+5 x^{4}-3 x^{2}+270$$
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