Example Question - polynomial
Here are examples of questions we've helped users solve.
Polynomial Expression Simplification
<p>First, group like terms in the expression.</p> <p>5u(2u) + 5u(3) - 4c => 10u^2 + 15u - 4c.</p> <p>Combine this with the second part of the expression:</p> <p>10u^2 + 15u - 8u - 12.</p> <p>This simplifies to:</p> <p>10u^2 + (15u - 8u) - 12 = 10u^2 + 7u - 12.</p> <p>The final simplified expression is:</p> <p>10u^2 + 7u - 12.</p>
Determining the Nature of Functions
<p>Given the function \( f(x) = x^2 - 5 \).</p> <p>To determine if the function has real roots, we can find the discriminant.</p> <p>The discriminant \( D \) is calculated as \( D = b^2 - 4ac \), where \( a = 1, b = 0, c = -5 \).</p> <p>Thus, \( D = 0^2 - 4 \cdot 1 \cdot (-5) = 20 \).</p> <p>Since \( D > 0 \), the function has two distinct real roots.</p> <p>To find the roots, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \).</p> <p>So, \( x = \frac{-0 \pm \sqrt{20}}{2 \cdot 1} = \frac{\pm 2\sqrt{5}}{2} = \pm \sqrt{5} \).</p> <p>Therefore, the roots are \( x = \sqrt{5} \) and \( x = -\sqrt{5} \).</p>
Determining Polynomial Expressions
<p>(a) \( x^2 - 5 \) is a quadratic expression because it can be written in the form \( ax^2 + bx + c \) with \( a = 1, b = 0, c = -5 \).</p> <p>(b) \( 2x^2 + x \) is a quadratic expression because it can be expressed as \( ax^2 + bx + c \) with \( a = 2, b = 1, c = 0 \).</p> <p>(c) \( 3x^2 - 3x + 1 \) is a quadratic expression since it conforms to the standard form \( ax^2 + bx + c \) with \( a = 3, b = -3, c = 1 \).</p> <p>(d) \( x^4 - 2x - 1 \) is not a quadratic expression as it has a degree of 4, which exceeds 2.</p>
Identifying Variable Types in Algebraic Expressions
<p>For the expression \( x^2 - 5 \):</p> <p>This is a polynomial expression where \( x \) is a variable.</p> <p>For the expression \( \frac{1}{m^2} \):</p> <p>This is a rational expression, where \( m \) is also a variable.</p>
Determine Variable Types
<p>For (a) \( x^2 - 5 \): This is a polynomial expression, and it contains a variable \( x \) which can take any real number value.</p> <p>For (b) \( \frac{1}{m^2} \): This expression also contains a variable \( m \), which denotes a non-zero real number since \( m \) cannot be equal to zero to avoid division by zero.</p>
Derivative of a Polynomial Expression
<p>Для того чтобы найти производную многочлена \( f(x) = 3x^2 + x^4 \), следует применить правило нахождения производной степенной функции: \((x^n)' = nx^{n-1}\).</p> <p>Тогда производная данного многочлена будет:</p> <p>\( f'(x) = (3x^2)' + (x^4)' \)</p> <p>\( f'(x) = 3 \cdot 2x^{2-1} + 4x^{4-1} \)</p> <p>\( f'(x) = 6x + 4x^3 \)</p>
Polynomial Simplification
<p>La ecuación proporcionada necesita ser simplificada. El proceso implica combinar términos semejantes y simplificar las fracciones. La ecuación está dada por:</p> <p>\[ \frac{{5a^3 - 20ab + 40b^3 - 30a + 20b}}{{8}} - \frac{{2a^3+40ab - 5b^3}}{{5}} \]</p> <p>Para simplificar esta expresión, multiplicamos cada término de la primera fracción por \( \frac{5}{5} \) y cada término de la segunda fracción por \( \frac{8}{8} \) para obtener denominadores comunes, que en este caso será 40.</p> <p>\[ \frac{{5(5a^3 - 20ab + 40b^3 - 30a + 20b)}}{{40}} - \frac{{8(2a^3+40ab - 5b^3)}}{{40}} \]</p> <p>Expandimos y simplificamos:</p> <p>\[ \frac{{25a^3 - 100ab + 200b^3 - 150a + 100b - 16a^3 - 320ab + 40b^3}}{{40}} \]</p> <p>Combinamos términos semejantes:</p> <p>\[ \frac{{(25a^3 - 16a^3) + (- 100ab - 320ab) + (200b^3 + 40b^3) - 150a + 100b}}{{40}} \]</p> <p>\[ \frac{{9a^3 - 420ab + 240b^3 - 150a + 100b}}{{40}} \]</p> <p>Finalmente, simplificamos los términos dividiendo cada uno por 40:</p> <p>\[ \frac{{9a^3}}{{40}} - \frac{{420ab}}{{40}} + \frac{{240b^3}}{{40}} - \frac{{150a}}{{40}} + \frac{{100b}}{{40}} \]</p> <p>\[ \frac{{9a^3}}{{40}} - 10.5ab + 6b^3 - \frac{{15a}}{{8}} + \frac{{5b}}{{2}} \]</p> <p>Y esta es la forma simplificada de la ecuación.</p>
Simplifying a Polynomial Expression
<p>Para simplificar la expresión proporcionada en la imagen, debemos combinar términos similares y restar los polinomios. La expresión es la siguiente:</p> <p>\[ \frac{5a^2 + 20ab + 4b^2 - 30a + 20b}{60} - \frac{a^2 + 6ab}{2} + \frac{4b}{5} \]</p> <p>Primero simplificamos cada término dividiendo por los denominadores apropiados:</p> <p>\[ \frac{5a^2}{60} + \frac{20ab}{60} + \frac{4b^2}{60} - \frac{30a}{60} + \frac{20b}{60} - \frac{a^2}{2} - \frac{6ab}{2} + \frac{4b}{5} \]</p> <p>Luego simplificamos más:</p> <p>\[ \frac{a^2}{12} + \frac{ab}{3} + \frac{b^2}{15} - \frac{a}{2} + \frac{b}{3} - \frac{a^2}{2} - 3ab + \frac{4b}{5} \]</p> <p>Ahora, combinamos términos semejantes:</p> <p>\[ \left( \frac{a^2}{12} - \frac{a^2}{2} \right) + \left( \frac{ab}{3} - 3ab \right) + \frac{b^2}{15} - \frac{a}{2} + \left( \frac{b}{3} + \frac{4b}{5} \right) \]</p> <p>Convertimos todos los términos a un denominador común para poder combinarlos fácilmente:</p> <p>\[ \left( \frac{2a^2 - 6a^2}{12} \right) + \left( \frac{ab - 9ab}{3} \right) + \frac{b^2}{15} - \frac{a}{2} + \left( \frac{5b + 12b}{15} \right) \]</p> <p>Simplificamos los términos:</p> <p>\[ -\frac{4a^2}{12} + \frac{-8ab}{3} + \frac{b^2}{15} - \frac{a}{2} + \frac{17b}{15} \]</p> <p>Reducimos los términos a su mínima expresión:</p> <p>\[ -\frac{a^2}{3} - \frac{8ab}{3} + \frac{b^2}{15} - \frac{a}{2} + \frac{17b}{15} \]</p> <p>El resultado final es una expresión simplificada que combina todos los términos dados.</p>
Algebraic Expression Involving Binomial Squares
<p>Die gegebene algebraische Ausdruck ist:</p> <p>\(4 \cdot (6a+1)^2 - 3 \cdot (4a-2)^2\)</p> <p>Um diesen Ausdruck zu vereinfachen, quadratieren wir zuerst beide Binome:</p> <p>\((6a+1)^2 = 36a^2 + 12a + 1\)</p> <p>\((4a-2)^2 = 16a^2 - 16a + 4\)</p> <p>Dann multiplizieren wir diese Quadrate mit den entsprechenden Koeffizienten:</p> <p>\(4 \cdot (36a^2 + 12a + 1) = 144a^2 + 48a + 4\)</p> <p>\(3 \cdot (16a^2 - 16a + 4) = 48a^2 - 48a + 12\)</p> <p>Jetzt subtrahieren wir den zweiten Ausdruck vom ersten:</p> <p>\(144a^2 + 48a + 4 - (48a^2 - 48a + 12)\)</p> <p>\(144a^2 + 48a + 4 - 48a^2 + 48a - 12\)</p> <p>Zusammenfassen der ähnlichen Terme ergibt:</p> <p>\((144a^2 - 48a^2) + (48a + 48a) + (4 - 12)\)</p> <p>\(96a^2 + 96a - 8\)</p> <p>Das ist die vereinfachte Form des gegebenen Ausdrucks.</p>
Polynomial Expression Simplification
Die gegebene Ausdruck lautet: <p> \(4 \cdot (6a + 1)^2 - 3 \cdot (4a - 2)^2 \) </p> Dies soll vereinfacht werden. Wir verwenden die binomischen Formeln \((x + y)^2 = x^2 + 2xy + y^2\) und \((x - y)^2 = x^2 - 2xy + y^2\). Beginnen wir mit dem Ausquadrieren der beiden Klammern: <p> \(= 4 \cdot ((6a)^2 + 2 \cdot 6a \cdot 1 + 1^2) - 3 \cdot ((4a)^2 - 2 \cdot 4a \cdot 2 + 2^2)\) </p> <p> \(= 4 \cdot (36a^2 + 12a + 1) - 3 \cdot (16a^2 - 16a + 4)\) </p> Nun multiplizieren wir die Koeffizienten mit den ausmultiplizierten Termen: <p> \(= 4 \cdot 36a^2 + 4 \cdot 12a + 4 \cdot 1 - 3 \cdot 16a^2 + 3 \cdot 16a - 3 \cdot 4\) </p> <p> \(= 144a^2 + 48a + 4 - 48a^2 + 48a - 12\) </p> Vereinfachen wir nun den Ausdruck, indem wir ähnliche Terme zusammenfassen: <p> \(= 144a^2 - 48a^2 + 48a + 48a + 4 - 12\) </p> <p> \(= 96a^2 + 96a - 8\) </p> Damit ist die vereinfachte Form des Ausdrucks: <p> \(96a^2 + 96a - 8\) </p>
Summation of a Polynomial Sequence
<p>\[ \sum_{i=3}^{6} (8i^2 + i) = (8(3)^2 + 3) + (8(4)^2 + 4) + (8(5)^2 + 5) + (8(6)^2 + 6) \]</p> <p>\[ = (72 + 3) + (128 + 4) + (200 + 5) + (288 + 6) \]</p> <p>\[ = 75 + 132 + 205 + 294 \]</p> <p>\[ = 606 + 294 \]</p> <p>\[ = 900 \]</p>
Calculating the Limit of a Polynomial Function
<p>Para calcular el límite de la función polinómica cuando \( x \) tiende a 1, simplemente sustituimos el valor de \( x \) en la función, ya que es un polinomio y no presenta ninguna indeterminación en \( x = 1 \).</p> <p>\[ \lim_{{x \to 1}} (x^3 - 3x^2 - 2x + 1) = (1)^3 - 3(1)^2 - 2(1) + 1 \]</p> <p>\[ = 1 - 3 - 2 + 1 \]</p> <p>\[ = -3 \]</p> <p>Por lo tanto, el límite de la función polinómica cuando \( x \) tiende a 1 es \(-3\).</p>
Calculating the Limit of a Cubic Polynomial
<p>\lim_{{x \to -1}} (x^3 - 3x^2 - 2x + 1)</p> <p>\text{Para calcular el límite, sustituimos } x = -1 \text{ en la polinomio.}</p> <p>(-1)^3 - 3(-1)^2 - 2(-1) + 1</p> <p>= -1 - 3(1) + 2 + 1</p> <p>= -1 - 3 + 2 + 1</p> <p>= -1</p>
Factoring Algebraic Expressions
<p>Phần lớn của bức ảnh bị cắt đi, tuy nhiên, dựa trên phần còn lại, ta thấy một bài tập phân tích đa thức thành nhân tử bằng phương pháp IPP. Ta sẽ giải phần b) như một ví dụ:</p> <p>\[ b) \quad 4x^3 - 4x^2 + 4x^2 - 5x + 2x - 2 \]</p> <p>Đầu tiên, ta nhóm các hạng tử:</p> <p>\[ (4x^3 - 4x^2) + (4x^2 - 5x) + (2x - 2) \]</p> <p>Sau đó, ta phân tích từng nhóm:</p> <p>\[ = 4x^2(x - 1) + x(4x - 5) + 2(x - 1) \]</p> <p>Nhận thấy rằng cả ba nhóm đều có nhân tử chung là \((x - 1)\), ta tiếp tục nhóm hạng tử:</p> <p>\[ = (4x^2 + x + 2)(x - 1) \]</p> <p>Kết quả cuối cùng của việc phân tích đa thức thành nhân tử cho phần b) là:</p> <p>\[ 4x^3 - 4x^2 - 5x + 2 = (4x^2 + x + 2)(x - 1) \]</p>
Polynomial Equation Analysis
<p>Bạn chưa cung cấp đầy đủ thông tin hoặc câu hỏi cụ thể cần giải quyết liên quan đến các phương trình đa thức B và D trong hình ảnh vì vậy tôi không thể đưa ra lời giải cụ thể. Nếu bạn cần tìm đạo hàm, tích phân, hoặc giải bất kỳ bài toán cụ thể nào liên quan đến B hoặc D, vui lòng cung cấp thông tin chi tiết hơn.</p>
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com