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Chapter 4: Problem 114

Factorise \(x^{4}+1\) as a product of two quadratic polynomials with real coefficients.

Short Answer

Expert verified
The factorization of \(x^4 + 1\) is \((x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1)\).

Step by step solution

01

Recognize the structure of the expression

The expression given is \(x^4 + 1\). Notice that it's not easily factorable using integer or simple factorization techniques like difference of squares. However, it can be factorized using a technique involving complex numbers or recognizing a hidden quadratic form.
02

Rewrite the expression using a known identity

We recognize that \(x^4 + 1\) can be expressed in a form involving a sum of squares. Use the identity: \(a^2 + b^2 = (a + bi)(a - bi)\). In this case, let \(a = x^2\) and \(b = 1\), so \(x^4 + 1 = (x^2 + i)(x^2 - i)\).
03

Multiply the complex terms to verify

Check the multiplication of the complex roots: \((x^2 + i)(x^2 - i) = x^4 + x^2i - x^2i - i^2 = x^4 - (-1) = x^4 + 1\). This verifies the factorization holds true.
04

Express each factor as a quadratic with real coefficients

Each complex factor can be expressed as a quadratic with real coefficients. Recognize that: \[x^2 + i = (x^2 + 1x + 1)(x^2 - 1x + 1)\] Here, realize that the complex factorization contributes to two forms: \( (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1) \). We'll use the form \((x^2 + \frac{1}{\sqrt{2}} x + 1)(x^2 - \frac{1}{\sqrt{2}} x + 1)\) to revert the equation to real-coefficient quadratics.
05

Confirm the factorization

Check your final factorization by expanding each product.Multiply the quadratic polynomials:\( (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1) = (x^2)^2 - (\sqrt{2}x)^2 + 1 \)This simplifies back to \( x^4 + 1 \), confirming the factorization.

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Key Concepts

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

Complex Numbers
In mathematics, complex numbers are an extension of the real numbers and are used in polynomial factorization among other areas. Standard complex numbers have the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).

A complex number, such as \(x^2 + i\), showcases how real and imaginary parts interact in polynomial equations. They aid in expressions that seem initially non-factorable within the realm of real numbers alone.

  • In the problem \(x^4 + 1\), expressing it as \(x^4 + (0)x^2 + 1 = (x^2 + i)(x^2 - i)\) demonstrates utilizing complex numbers for factorization.
  • Each factor \(x^2 + i\) and \(x^2 - i\) represents a blend of polynomial and imaginary units which simplifies balancing polynomials across dimensions even if they do not appear explicitly in real form.
Understanding complex numbers enables recognizing unique factorization opportunities in algebra, especially when traditional real-based methods falter.
Quadratic Equations
Quadratic equations are polynomial equations of degree two. They have the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constant coefficients and \(a eq 0\).

The quadratic expression is considered the cornerstone of polynomials, being both a fundamental building block and a tool for factorization in higher degree polynomials as demonstrated in this exercise.

When addressing a factorization problem like \(x^4 + 1\), the task involves identifying simpler quadratic forms as components.

  • The given polynomial was adjusted to fit into a form of \(a^2 + b^2 = (a + bi)(a - bi)\), with \(a = x^2\) and \(b = 1\).
  • This identity essentially boiled down to creating two quadratic expressions by working with the squares and bridging them through multiplication: \((x^2 + \frac{1}{\sqrt{2}}x + 1)(x^2 - \frac{1}{\sqrt{2}}x + 1)\).
This strategic approach transcribes higher degree polynomials into manageable quadratic pieces, enabling simpler analysis and solution paths.
Real Coefficients
Polynomials with real coefficients are those wherein all terms and constants are real numbers without imaginary components.

In factorization, retaining real coefficients is crucial for ensuring solutions remain in the realm of real numbers whenever possible. In complex factorization scenarios, the goal often revolves around converting complex expressions into an equivalent real-coefficient form.

  • The exercise demonstrated transforming the complex product \((x^2 + i)(x^2 - i)\) into polynomials \((x^2 + \frac{1}{\sqrt{2}} x + 1)\) and \((x^2 - \frac{1}{\sqrt{2}} x + 1)\), both containing real coefficients.
  • This transformation maintains the integrity of real-number solutions while integrating necessary methods from complex numbers to branch out solution boundaries.
Working with real coefficients simplifies the real-number problem-solving, particularly in applications like graphs and real-world modeling where the presence of real numbers is pivotal.

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Most popular questions from this chapter

(i) the quotient and the remainder when we divide \(x^{10}+1\) by \(x^{3}-1\) (ii) the remainder when we divide \(x^{2013}+1\) by \(x^{2}-1\) (iii) the quotient and the remainder when we divide \(x^{m}+1\) by \(x^{n}-1,\) for \(m>n \geqslant 1\)

(a) Mark on the coordinate line all those points \(x\) for which two of the following inequalities are true, and five are false: $$ |x|>1,|x|>2,|x|>3,|x|>4,|x|>5,|x|>6,|x|>7 $$ (b) Mark on the coordinate line all those points \(x\) for which two of the following inequalities are true, and five are false: $$ |x-1|>1,|x-2|>2,|x-3|>3,|x-4|>4,|x-5|>5,|x-6|>6,|x-7|>7 $$

Complex numbers \(a+b i\), where both \(a\) and \(b\) are integers, are called Gaussian integers. Try to formulate a version of the "division algorithm" for "division with remainder" (where the remainder is always "less than" the divisor in some sense) for pairs of Gaussian integers. Extend this to construct a version of the Euclidean algorithm to find the HCF of two given Gaussian integers. It is a profoundly erroneous truism ... that we should cultivate the habit of thinking what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Alfred North Whitehead \((1861-1947)\)

(a) Given two complex numbers in polar form: $$ w=r(\cos \theta+i \sin \theta), z=s(\cos \phi+i \sin \phi) $$ show that their product is precisely $$ w z=r s(\cos (\theta+\phi)+i \sin (\theta+\phi)) $$ (b) (de Moivre's Theorem: Abraham de Moivre \((1667-1754))\) Prove that $$ (\cos \theta+i \sin \theta)^{n}=\cos (n \theta)+i \sin (n \theta) $$ (c) Prove that, if $$ z=r(\cos \theta+i \sin \theta) $$ satisfies \(z^{n}=1\) for some integer \(n,\) then \(r=1\) The last three problems in this subsection look more closely at "roots of unity" \(-\) that is, roots of the polynomial equation \(x^{n}=1 .\) In the real domain, we know that: (i) when \(n\) is odd, the equation \(x^{n}=1\) has exactly one root, namely \(x=1 ;\) and (ii) when \(n\) is even, the equation \(x^{n}=1\) has just two solutions, namely \(x=\pm 1\). In contrast, in the complex domain, there are \(n " n^{\text {th }}\) roots of unity" . Problem \(\mathbf{1 3 0}(\mathrm{c})\) shows that these "roots of unity" all lie on the unit circle, centered at the origin. And if we put \(n \theta=2 k \pi\) in Problem \(130(\mathrm{~b})\) we see that the \(n n^{\text {th }}\) roots of unity include the point \(" 1=\cos 0+i \sin 0\) ", and are then equally spaced around that circle with \(\theta=\frac{2 k \pi}{n}(1 \leqslant k \leqslant n-1),\) and form the vertices of a regular \(n\) -gon.

(a) Suppose you know the position vectors \(\mathbf{p}, \mathbf{q}, \mathbf{r}\) corresponding to the midpoints of the three sides of a triangle. Can you reconstruct the vectors \(\mathbf{x}, \mathbf{y}, \mathbf{z}\) corresponding to the three vertices? (b) Suppose you know the vectors \(\mathbf{p}, \mathbf{q}, \mathbf{r},\) s corresponding to the midpoints of the four sides of a quadrilateral. Can you reconstruct the vectors \(\mathbf{w}, \mathbf{x}\) \(\mathbf{y}, \mathbf{z}\) corresponding to the four vertices? (c) Suppose you know the vectors \(\mathbf{p}, \mathbf{q}, \mathbf{r}, \mathbf{s}, \mathbf{t}\) corresponding to the midpoints of the five sides of a pentagon. Can you reconstruct the vectors \(\mathbf{v}, \mathbf{w}, \mathbf{x},\) y, z corresponding to the five vertices? \(\triangle\) The previous five problems explore a common structural theme \(-\) namely the link between certain sums (or averages) and the original, possibly unknown, data. However this algebraic link was in every case embedded in some practical, or geometrical, context. The next few problems have been stripped of any context, leaving us free to focus on the underlying structure in a purely algebraic, or arithmetical, spirit.
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