Polynomial And Rational Functions Vocabulary

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1. What is a Quadratic Inequality?

A linear equation with two variables.

An inequality that involves a quadratic polynomial

An equation with fractions as coefficients.

A trigonometric equation.

A quadratic inequality is specifically related to quadratic polynomials and not linear equations, equations with fractions, or trigonometric equations.

Explanation

A quadratic inequality is specifically related to quadratic polynomials and not linear equations, equations with fractions, or trigonometric equations.

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About This Quiz

Explore the essentials of polynomial and rational functions, focusing on their properties, operations, and applications. This educational content is designed to enhance understanding of key mathematical concepts, suitable... see morefor students aiming to master advanced algebraic techniques. see less

2. Theorem on Bounds

If c > 0 and all terms in the bottom row are negative, then c is an upper bound for the roots of P(x) = 0.

If c = 0 and all terms in the bottom row are zero, then c is a root of P(x) = 0.

Suppose that P(x) is a polynomial with real coefficients and a positive leading coefficient and synthetic division with x - c is performed.If c > 0 and all terms in the bottom row are nonnegative, then c is an upper bound for the roots of P(x) = 0If c < 0 and the terms in the bottom row alternate in sign, the c is a lower bound for the roots of P(x) = 0

If c < 0 and all terms in the bottom row are nonnegative, then c is a lower bound for the roots of P(x) = 0.

The correct answer provides conditions for c to be an upper or lower bound for the roots of the polynomial P(x) = 0 based on synthetic division results. The incorrect answers provide alternative scenarios that do not align with the given conditions.

Explanation

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3. What is an asymptote?

Vertical or horizontal line which the curve gets closer and closer to but never touches

A point where the curve intersects the x-axis.

A solution to an equation where both variables are equal to zero.

A curve that intersects with another curve.

An asymptote is a line that a curve approaches but never touches as it extends to infinity. It can be vertical or horizontal, representing limits in the curve's behavior.

Explanation

An asymptote is a line that a curve approaches but never touches as it extends to infinity. It can be vertical or horizontal, representing limits in the curve's behavior.

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4. What does it mean when the statement 'y varies directly as x'?

Direct variation indicates that y = x^2 for some fixed constant.

Direct variation implies that y is inversely proportional to x in all cases.

The statement y varies directly as x means that y = kx for some fixed nonzero real number k. Direct variation is a situation in which as one quantity increases, the other increases at a constant rate.

Direct variation means that y and x are completely unrelated and have no correlation.

Direct variation specifically refers to the relationship where one quantity (y) is directly proportional to another quantity (x) by a constant factor (k). This means that as x increases or decreases, y will increase or decrease at a fixed rate, which is captured in the equation y = kx.

Explanation

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5. What is the vertex form of a quadratic function?

F(x) = ax^2 + bx + c

F(x) = a(x - h)2 + k

F(x) = a(x + h)^2 - k

F(x) = a(x + h)(x - k)

The vertex form of a quadratic function is represented as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.

Explanation

The vertex form of a quadratic function is represented as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.

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6. Factor Theorem.

The number c is a zero of the polynomial function y = P(x) if and only if x - c is a facor of the polynomial P(x)

Factor theorem applies only to linear functions.

Zero of a polynomial function is always the coefficient of the highest degree term.

X - c is not related to the zero of a polynomial function.

The factor theorem states that a number is a zero of a polynomial function if and only if the corresponding linear factor (x - c) is a factor of the polynomial. This theorem is fundamental in algebra and helps in finding roots of polynomial equations.

Explanation

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7. What is the Rational Zero Theorem?

B) The Rational Zero Theorem states that the leading coefficient of a polynomial function can never be a factor of any rational zero.

If f(x) is a polynomial function with integral coefficients and p/q is a rational zero of f(x) the p is a factor of the constant term and q is a factor of the leading coefficient

A) The Rational Zero Theorem states that only irrational numbers can be solutions to polynomial functions with integral coefficients.

C) The Rational Zero Theorem states that the constant term of a polynomial function does not affect the rational zeros of the function.

The Rational Zero Theorem helps in simplifying the process of finding rational zeros of a polynomial function by defining the relationship between the zeros and the coefficients of the function.

Explanation

The Rational Zero Theorem helps in simplifying the process of finding rational zeros of a polynomial function by defining the relationship between the zeros and the coefficients of the function.

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8. What is Descartes's Rule of Signs?

The number of negative real roots of the equation is always equal to the number of sign variations of P(-x).

P(x) is not required to be a polynomial equation with real coefficients for Descartes's Rule of Signs to apply.

Suppose P(x) = 0 is a polynomial equation with real coefficients and with terms written in descending order. Then,the number of positive real roots of the equation is either equal to the number of sign variation of P(x) or is less than that by an even integerthe number of negative real roots ot he equation is either equal to the number of sign variations of P(-x) or is less than that by an even integer

The number of positive real roots is always greater than the number of sign variations of P(x).

Descartes's Rule of Signs provides a way to determine the potential number of positive and negative real roots in a polynomial equation with real coefficients by examining the sign variations.

Explanation

Descartes's Rule of Signs provides a way to determine the potential number of positive and negative real roots in a polynomial equation with real coefficients by examining the sign variations.

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9. What does it mean when it is said that y varies inversely as x?

Y equals X

The statement y varies inversely as x means that y = for some fixed nonzero real number k.Inverse variation is a situation in which as one quantity increases, the other decreases at a constant rate.

Y equals X squared

Y equals kX

Inverse variation is a relationship between two variables in which one variable increases as the other decreases at a constant rate. The correct answer states the correct formula for inverse variation where y varies inversely as x.

Explanation

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10. What is the standard form of a quadratic function?

F(x) = ax^2 - bx + c

F(x) = ax^3 + bx + c

F(x) = x^2 + bx + c

F(x) = ax2 + bx + c

The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.

Explanation

The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.

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11. What is the Axis of Symmetry?

Vertical line over which a parabola is symmetric

Horizontal line where a parabola intersects the x-axis.

Diagonal line that cuts through a parabola.

Curved line that connects the highest and lowest points of a parabola.

The Axis of Symmetry of a parabola is a vertical line that divides the parabola into two equal halves, ensuring symmetry.

Explanation

The Axis of Symmetry of a parabola is a vertical line that divides the parabola into two equal halves, ensuring symmetry.

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12. Remainder Theorem

The Remainder Theorem states that the remainder when a polynomial is divided by x - c is always 0.

The Remainder Theorem only applies to linear polynomials.

The Remainder Theorem states that the remainder when a polynomial is divided by x + c is P(c).

If R is th eremainder when a polynomial P(x) is divided by x - c, the R = P(c)

The correct answer explains that in the Remainder Theorem, the remainder (R) when a polynomial P(x) is divided by x - c is equal to P(c). This helps in determining the value of the polynomial at a specific point and understanding the behavior of polynomials.

Explanation

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13. What is Synthetic Division?

A technique used in computer programming to generate fake data.

A mathematical operation involving mixing two synthetic numbers.

An algorithm for dividing a polynomial by (x + a) or (x - a)

A process of creating artificial compounds in a laboratory setting.

Synthetic Division is a mathematical algorithm specifically used for polynomial division by linear binomials in the form of (x + a) or (x - a). It is a systematic and efficient method to simplify polynomial division operations.

Explanation

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14. What does multiplicity refer to in the context of polynomials?

The sum of all terms in a polynomial.

The process of finding the roots of a polynomial.

The number of times a root occurs in the complete factorization of a polynomial

The degree of the leading term in a polynomial.

The correct answer defines multiplicity as the frequency of a root in the factorization of a polynomial, which is distinct from the other options provided.

Explanation

The correct answer defines multiplicity as the frequency of a root in the factorization of a polynomial, which is distinct from the other options provided.

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15. What is a rational function?

A function that cannot be expressed with a finite number of terms.

A function that has a graph with no asymptotes or holes.

A function of the form where P(x) and Q(x) are polynomials