Algebra: Quadratic Equations and Polynomials

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Questions and Answers

What does the expression $b² - 4ac$ represent in the context of a quadratic equation?

The coefficient of x

The product of the roots

The sum of the roots

The discriminant (correct)

If the discriminant $D$ is greater than zero, what can be concluded about the roots of the quadratic equation?

The roots are non-real complex

The roots are real and distinct (correct)

There are no roots

The roots are equal

Using the Remainder Theorem, what is the remainder when the polynomial $f(x) = x² - 3x - x - 6$ is divided by $x + 3$?

51 (correct)

Which of the following statements is TRUE about the roots of the quadratic equation $x² - x + i = 0$?

The roots are complex conjugates Signup and view all the answers

What is the factor form of a quadratic equation where $x₁$ and $x₂$ are its roots?

$a(x - x₁)(x - x₂)$ Signup and view all the answers

If $f(a) = 0$, what does it imply regarding the polynomial $f(x)$?

$x - a$ is a factor of $f(x)$ Signup and view all the answers

In the expression $x² - 2x - 1 = 0$, what does the value $D = 8$ indicate about the roots?

The roots are real and distinct Signup and view all the answers

Which of the following values of $D$ indicates that the roots of the equation are completely non-real?

$D < 0$ Signup and view all the answers

What is the degree of the polynomial f(x) = 2x³ + 3x² - 2x - 5?

3 Signup and view all the answers

Which of the following expressions represents the factor theorem for a polynomial f(x)?

If f(a) = 0, then (x - a) is a factor of f(x) Signup and view all the answers

What is the remainder when dividing f(x) = 2x³ + 3x² - 2x - 5 by g(x) = x - 1?

-5 Signup and view all the answers

What does the fundamental theorem of algebra state about polynomials of degree n?

They have n roots, counting multiplicities. Signup and view all the answers

If Q(x) is a polynomial of degree n-1 and has a root α, what can be inferred about the polynomial f(x) if f(x) = (x - α) Q(x)?

f(x) is a polynomial of degree n. Signup and view all the answers

Which expression correctly shows the factorization of f(x) when it has roots α₁, α₂, and αₙ?

f(x) = (x - α₁)(x - α₂)...(x - αₙ) Signup and view all the answers

In the polynomial division of f(x) by g(x), how does the degree of the quotient relate to the degrees of f(x) and g(x)?

It can be equal to the degree of f(x) minus the degree of g(x). Signup and view all the answers

What is implied by the statement f(x) = 0 if f(x) has n distinct roots?

It can be factored into (x - α₁)(x - α₂)...(x - αₙ). Signup and view all the answers

What is the equation derived from the roots 2, 2 + √3, and 2 - √3?

x³ - 2x² + 8x = 0 Signup and view all the answers

Which of the following statements about irrational roots is true?

If an equation has an irrational root, it must also have its conjugate as a root. Signup and view all the answers

What is the result of multiplying (1 + 3i) and (1 - 3i)?

10 Signup and view all the answers

According to the theorem, complex roots of a polynomial with real coefficients occur in what manner?

In conjugate pairs. Signup and view all the answers

What is the degree of the polynomial derived from the roots 1, 2, and 3?

3 Signup and view all the answers

What is the final simplified form of the equation derived from the expression (x - (1 + 3i))(x - (1 - 3i)) = 0?

x² - 2x + 10 = 0 Signup and view all the answers

Given the equation x² + 4x + 8 = 0, which of the following statements is true?

Its upper limit to real roots is greater than 2. Signup and view all the answers

In the context of finding an upper limit for real roots, what values represent 'k' and 'G' in the equation x⁵ + 4x² - 7x² - 40x + 1 = 0?

k = 3, G = 40 Signup and view all the answers

According to the theorem related to the coefficients of a real equation, what is the expression used to determine an upper limit when necessary?

1 + k√(G/a₀) Signup and view all the answers

Which of the following statements best describes an upper limit to the real roots of an equation with no negative coefficients?

Any absolute number greater than the maximum real root serves as an upper limit. Signup and view all the answers

What does the expression $f(x) = x^4 - 9x^3 + 24x^2 - 23x + 15$ demonstrate regarding its roots?

3 is a root of the polynomial equation. Signup and view all the answers

How is the complex number represented in polar form?

z = r(cosθ + isinθ) Signup and view all the answers

Which of the following statements about complex numbers is true?

The general form of a complex number is $z = x + iy$. Signup and view all the answers

What is the value of $k$ when applying the upper limit $u.l.$ based on the provided conditions?

1 Signup and view all the answers

If $r^2 + y^2 = r^2$, what can be inferred about $x$ and $y$?

This equality indicates a relationship between x and y in polar coordinates. Signup and view all the answers

Which of the following numbers is a possible value of $c$ in the polynomial root expression $x^4 - 8x^3 - 21x^2 + 22x + c = 0$?

40 Signup and view all the answers

When using synthetic division for the polynomial $x^4 - 9x^3 + 24x^2 -23x + 15$, what does a result of '0' indicate?

The divisor is a factor of the polynomial. Signup and view all the answers

Which of the following equations represents the quadratic part of the polynomial when the highest degree term is removed?

aₙ₋₂x² + aₙ₋₁x + aₙ = 0 Signup and view all the answers

Study Notes

Quadratic Equation

The general form of a quadratic equation is ax² + bx + c = 0 where a ≠ 0.
By completing the square, the quadratic formula can be derived: x = [-b ± √(b² - 4ac)]/2a.
The discriminant, D = b² - 4ac, determines the nature of the roots:
- If D = 0, the roots are real and equal (perfect square).
- If D > 0, the roots are real and distinct.
- If D < 0, the roots are complex conjugates.
The sum of the roots (x₁ + x₂) is -b/a and the product of the roots (x₁.x₂) is c/a.
The quadratic equation can be factored as a(x - x₁)(x - x₂) = 0.

Polynomials

A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
The general form of a polynomial is f(x) = axⁿ + bxⁿ⁻¹ + ... + c₁x + c₀, where n is the degree of the polynomial.

Remainder Theorem

When a polynomial f(x) is divided by (x - a), the remainder is f(a).
If the remainder is 0, then (x - a) is a factor of f(x).

Factor Theorem

If f(a) = 0, then (x - a) is a factor of the polynomial f(x).

Theorem of Root of Equation

This theorem provides an alternative to the division method for linear divisors only.
By applying synthetic division, a polynomial can be rewritten in the form f(x) = q(x) + r, where q(x) is the quotient and r is the remainder.

Factor Form of a Polynomial

A polynomial of degree n can be factorized as f(x) = (x - a₁)(x - a₂) ... (x - aₙ)aₙ, where a₁, a₂, ..., aₙ are its roots.

Theorem on Roots of Equation

An equation of degree 'n' has exactly 'n' roots, considering multiplicity.
Each root can be expressed as (x - αᵢ)aₙ where αᵢ is the root and aₙ is a constant.

Fundamental Theorem of Algebra

Every algebraic equation with complex coefficients has at least one complex root. As a consequence, each polynomial of degree 'n' has precisely 'n' roots (counting multiplicity).

Theorem on Irrational Roots

Irrational roots of an algebraic equation with rational coefficients always occur in conjugate pairs.
If a + √b is a root, then its conjugate, a - √b, is also a root.

Theorem on Complex Roots

Complex roots of an equation with real coefficients always occur in conjugate pairs.
If a + ib is a root, then its conjugate, a - ib, is also a root.

Upper Limit to Real Roots of Equation

A number γ is an upper limit to the real roots of a real equation if it exceeds all the real roots.
An equation with no negative coefficients has any positive number as an upper limit.

Theorem on Finding Upper Limits of Real Roots

If the first positive coefficient in a real equation is preceded by 'k' zero coefficients and 'G' is the greatest of the numerical values of the negative coefficients, then each real root is less than 1 + k√(G/a₀).

Newton's Method for Integral Roots

This method helps determine if a number is a root of a polynomial.
It involves dividing the polynomial by the given number and checking if the remainder is 0.
If the remainder is 0, the number is a root.

Complex Numbers

Complex numbers are extensions of real numbers, including the imaginary unit 'i' where i = √-1.
A general complex number is represented as z = x + iy, where x and y are real numbers.
Complex numbers can also be expressed in polar form: z = r(cosθ + isinθ), where r is the magnitude and θ is the angle.
Complex numbers are denoted by 'C'.
The set of all complex numbers is represented by 'C'.

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Description

Test your understanding of quadratic equations and polynomials through this quiz. It covers key concepts such as the quadratic formula, the discriminant, and the Remainder Theorem. Challenge yourself to apply these principles to various mathematical problems.