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Explore Discriminant through 10+ example sentences from English, with an explanation of the meaning. Ideal for language learners, writers and word enthusiasts.

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Discriminant in a sentence

Discriminant meaning

  1. An expression that gives information about the roots of a polynomial; for example, the expression D = b² - 4ac determines whether the roots of the quadratic equation ax² + bx + c = 0 are real and distinct (D > 0), real and equal (D = 0) or complex (D < 0).
  2. The invariant (on the vector space of forms of degree d in n variables) that vanishes exactly when the corresponding hypersurface in Pⁿ⁻¹ is singular.

Using Discriminant

  • The main meaning on this page is: An expression that gives information about the roots of a polynomial; for example, the expression D = b² - 4ac determines whether the roots of the quadratic equation ax² + bx + c = 0 are real and distinct (D > 0), real and equal (D = 0) or complex (D < 0). | The invariant (on the vector space of forms of degree d in n variables) that vanishes exactly when the corresponding hypersurface in Pⁿ⁻¹ is singular.
  • In the example corpus, discriminant often appears in combinations such as: the discriminant, discriminant is, discriminant of.

Context around Discriminant

  • Average sentence length in these examples: 26.6 words
  • Position in the sentence: 8 start, 10 middle, 2 end
  • Sentence types: 20 statements, 0 questions, 0 exclamations

Often before:

fundamental 1×

Often after:

equals 1× making 1× determines 1× gives 1×

Sentence patterns with Discriminant

These short patterns come directly from the example sentences and show how the word is often embedded:

if the discriminant is non 2× 1 the discriminant equals the 1× a fundamental discriminant if it 1× as the discriminant is a 1× calculating the discriminant the curve 1× case the discriminant determines the 1× d its discriminant 1× degrees the discriminant is always 1×

Sentence openings and endings

Common sentence openings:

a second form 1× algebraically this involves 1× as shown in 1× as the discriminant 1× definition in terms 1× for example in 1×

Common sentence endings:

a quadratic form 1× case is 368 1× correct significant figures 1× d its discriminant 1× dropped for simplicity 1× equal to zero 1×

Words around Discriminant

Just before it in the word index:

discrimi 2 discribed 2 discretizing 2 discretized 6 discretization 8 discretised 2

Just after it in the word index:

discriminate 227 discriminated 204 discriminates 37 discriminating 121 discrimination 2.134 discriminations 9

Similar in corpus frequency:

aberrant 31 abloh 31 absurdities 31 acceptor 31 acetaminophen 31 achaemenid 31

Corpus analysis for Discriminant

  • In this selection, "discriminant" usually appears in the middle of the sentence. The average example has 26.6 words, and this corpus slice is mostly made up of statements.
  • Around the word, fundamental, equals, making and determines stand out and add context to how "discriminant" is used.
  • Recognizable usage signals include if the discriminant is non and 1 the discriminant equals the. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "discriminant" sits close to words such as aberrant, abloh and absurdities, which helps place it inside the broader word index.

Example types with discriminant

The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:

Short example sentences

Moreover, in general not all possible terms will occur in the discriminant. (12 words)

In this case the discriminant determines the number and nature of the roots. (13 words)

Let f be a polynomial with coefficients in a commutative ring A and D its discriminant. (16 words)

Longer example sentences

As the discriminant is a symmetric function in the roots, it can also be expressed in terms of the coefficients of the polynomial, since the coefficients are the elementary symmetric polynomials in the roots; such a formula is given below. (40 words)

Nature of the roots The discriminant gives additional information on the nature of the roots beyond simply whether there are any repeated roots: for polynomials with real coefficients, it also gives information on whether the roots are real or complex. (40 words)

However, the discriminant in a PDE is given by due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is with the factor of 4 dropped for simplicity. (39 words)

Example sentences (20)

Algebraically, this involves calculating the discriminant : The curve is non-singular if and only if the discriminant is not equal to zero.

However, the discriminant in a PDE is given by due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is with the factor of 4 dropped for simplicity.

Landau, p. 168, credits Gauss as well as Dirichlet An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field.

The discriminant of the cubic polynomial : is : For higher degrees, the discriminant is always a polynomial function of the coefficients.

Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

A second form of cancellation can occur between the terms b 2 and 4ac of the discriminant, which can lead to loss of up to half of correct significant figures.

As shown in Figure 3, if the discriminant is positive, the graph touches the x -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x -axis.

As the discriminant is a symmetric function in the roots, it can also be expressed in terms of the coefficients of the polynomial, since the coefficients are the elementary symmetric polynomials in the roots; such a formula is given below.

Advertentie

Definition In terms of the roots, the discriminant is given by : where is the leading coefficient and are the roots (counting multiplicity ) of the polynomial in some splitting field.

For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368.

Further, for a quadratic polynomial with rational coefficients, it factors over the rationals if and only if the discriminant – which is necessarily a rational number, being a polynomial in the coefficients – is in fact a square.

If the discriminant were positive, the distance would be non-zero, and there will be two solutions, as expected.

If the lead coefficient a equals 1, the discriminant equals the product of the squared differences of all pairs of roots of the polynomial.

If the quantity under the square root ( the discriminant ) is negative, then the ray does not intersect the sphere.

In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.

In this case the discriminant determines the number and nature of the roots.

Let f be a polynomial with coefficients in a commutative ring A and D its discriminant.

Moreover, in general not all possible terms will occur in the discriminant.

Nature of the roots The discriminant gives additional information on the nature of the roots beyond simply whether there are any repeated roots: for polynomials with real coefficients, it also gives information on whether the roots are real or complex.

Suppose that U has even dimension and a non-singular bilinear form with discriminant d, and suppose that V is another vector space with a quadratic form.

Advertentie

Common combinations with discriminant

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "discriminant" in a sentence?
An example: "Algebraically, this involves calculating the discriminant : The curve is non-singular if and only if the discriminant is not equal to zero." This page contains 10+ example sentences with the word "discriminant" from authentic English texts.
What does "discriminant" mean?
Discriminant means: An expression that gives information about the roots of a polynomial; for example, the expression D = b² - 4ac determines whether the roots of the quadratic equation ax² + bx + c = 0 are real and distinct (D > 0), real and equal (D = 0) or complex (D < 0).
How many example sentences with "discriminant" are there?
Voorbeeldzinnen.info contains at least 10+ example sentences with "discriminant", drawn from a database of millions of English sentences.