Example Question - polynomials
Here are examples of questions we've helped users solve.
Simplifying a Rational Expression Involving Polynomials
<p> \( y = \frac{x}{(2x-1) \cdot (3) \cdot (x-1)} \) </p> <p> To simplify the expression, factor out and reduce common terms if any. </p> <p> No common terms to factor out or reduce. </p> <p> The expression is already in its simplest form. So, the simplified expression is: </p> <p> \( y = \frac{x}{6x^2 - 3x - 2x + 1} \) </p> <p> \( y = \frac{x}{6x^2 - 5x + 1} \) </p>
Simplifying an Algebraic Expression Involving Fractions and Polynomials
<p>We need to simplify the given algebraic expression:</p> <p>\[\frac{11 + x}{x^3} + 2x(5 - x)\]</p> <p>First, distribute \(2x\) across the parenthesis:</p> <p>\[2x \cdot 5 - 2x \cdot x\]</p> <p>\[10x - 2x^2\]</p> <p>The expression becomes:</p> <p>\[\frac{11 + x}{x^3} + 10x - 2x^2\]</p> <p>Since there is no common denominator, the expression is already simplified. However, if the problem requires combining like terms or a common denominator for some reason, additional context would be needed.</p>
Factoring Polynomial Expressions
Chúng ta hãy giải quyết bài toán số \( 2 \): <p>\( (x+1)(x+2)(x+3)(x+4) - 24 \)</p> <p>Bước 1: Nhận ra \( (x+1)(x+4) \) và \( (x+2)(x+3) \) là hai cặp số hạng liền kề của một dãy số liên tiếp.</p> <p>\( (x+1)(x+4) = x^2 + 5x + 4 \)</p> <p>\( (x+2)(x+3) = x^2 + 5x + 6 \)</p> <p>Bước 2: Nhân hai biểu thức trên.</p> <p>\( (x^2 + 5x + 4)(x^2 + 5x + 6) \)</p> <p>Vì cách làm trên khá dài và phức tạp, ta thử nhận ra một mẫu số chuẩn hóa:</p> <p>Bước 3: Phát hiện \( (x^2 + 5x + 4)(x^2 + 5x + 6) \) gần giống \( (x^2 + 5x + 5)^2 \), nhưng cần trừ đi \( 1 \).</p> <p>\( (x^2 + 5x + 4)(x^2 + 5x + 6) = (x^2 + 5x + 5)^2 - 1^2 \)</p> <p>Bước 4: Sử dụng công thức hiệu của hai bình phương.</p> <p>\( (x^2 + 5x + 5 + 1)(x^2 + 5x + 5 - 1) \)</p> <p>\( (x^2 + 5x + 6)(x^2 + 5x + 4) \)</p> <p>Bước 5: So sánh với biểu thức gốc và điều chỉnh để có 24 ở cuối.</p> <p>\( ((x^2 + 5x + 6)(x^2 + 5x + 4)) - 24 \)</p> <p>Bước 6: Đặt \( A = x^2 + 5x + 6 \) và \( B = x^2 + 5x + 4 \) để dễ quản lý.</p> <p>\( (AB) - 24 = (A - 2)(B + 2) \)</p> <p>Đây là kết quả sau khi đã nhân và thực hiện phép trừ.</p>
Simplifying Algebraic Expression
3p(p - q) - (2p - q)² 3p² - 3pq - (4p² - 4pq + q²) 3p² - 3pq - 4p² + 4pq - q² -p² + pq - q²
Finding the Least Common Multiple of Monomials and Polynomials
To find the least common multiple (LCM) of the given monomials \( 5x^7 \), \( 5x^6 - 10x^5 \), and \( 2x^8 + 8x^7 + 8x^6 \), you would first factor each monomial or polynomial into prime factors and include variables with their highest powers. Let's start by factoring the given expressions: 1. \( 5x^7 \) is already in its factored form. 2. \( 5x^6 - 10x^5 \) can be factored as \( 5x^5(x - 2) \). 3. \( 2x^8 + 8x^7 + 8x^6 \) can be factored by pulling out the greatest common factor, which is \( 2x^6 \), leaving us with \( 2x^6(x^2 + 4x + 4) \). However, since we are only interested in the LCM, which concerns the greatest powers of variables and prime factors, we don't need to factor the quadratic expression further. Now, we take the highest powers of each prime factor found in the terms: - The highest power of 2 is \( 2^1 \) from \( 2x^6(x^2 + 4x + 4) \). - The highest power of 5 is \( 5^1 \) from \( 5x^7 \) and \( 5x^5(x - 2) \). - The highest power of \( x \) is \( x^8 \) from \( 2x^8 + 8x^7 + 8x^6 \). Combining these, the LCM of the given expressions is: \[ 2^1 \cdot 5^1 \cdot x^8 = 10x^8 \] Therefore, the LCM is \( 10x^8 \).
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