🧮 algebra

**Problem Set 3.2: Addition and Subtraction of Polynomials** 1. Add $(3m - 5k - h) + (-6m + 4k - 5h)$

1. Let's state the problem: Solve the inequality $$\frac{2 - x}{2} > -1$$. 2. Multiply both sides by 2 (which is positive, so inequality direction remains the same):

1. **Problem Statement:** We are given a sequence defined by the general term $$x_n = \frac{5n + 6}{5n}$$ and asked to analyze it. 2. **Simplify the expression:**

1. Simplify the expression $$\frac{2^{10} \cdot 3^{15}}{9 \cdot 3^{10} \cdot 12}$$ - Rewrite the bases: $$9 = 3^2, \quad 12 = 2^2 \cdot 3$$

1. Simplify $2^4 \cdot 2^{-1}$. Using the exponent rule $a^m \cdot a^n = a^{m+n}$:

1. **State the problem:** Kialani sells orchids and lilies at different prices. We know:

1. Diketahui fungsi $f(x) = \frac{|9-3x|}{x-3}$. 2. Kita akan meninjau limit $f(x)$ ketika $x \to 3$ dari kiri ($3-$) dan dari kanan ($3+$).

1. The problem is to identify coefficients $a$, $b$, and $c$ for each quadratic equation. 2. For $x^2 + x - 6 = 0$, rewrite the equation as is: $a=1$, $b=1$, $c=-6$.

1. Diberikan fungsi pecahan \(f(x) = \begin{cases} x^2 - a, & x < 2 \\ x + a, & x > 2 \end{cases}\). Kita diminta menentukan nilai \(a\) agar limit \(\lim_{x \to 2} f(x)\) ada. 2.

1. The problem is to solve the quadratic equation $$x^2 - 2x - 15 = 0$$. 2. To solve this quadratic, we can factor the quadratic expression on the left side.

1. The problem is to solve the quadratic equation $$2x^2 - 6x + 1 = 0$$. 2. We will use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = 2$, $b = -6$, a

1. State the problem: Solve the equation $$2x - 3 = \frac{3x - 5}{4}$$ for $x$. 2. Eliminate the denominator by multiplying both sides of the equation by 4:

1. Mukhang may nais kang linawin tungkol sa bilang ng mga termino sa isang mathematical expression. 2. Karaniwan, kapag sinabing "6 term," ibig sabihin nito ay may anim na magkahiw

1. State the problem: Solve the quadratic equation $$2x^2 = 1 - 2x$$. 2. Rearrange all terms to one side to set the equation to 0:

1. **State the problem:** We are given a recurrence relation

1. State the problem: Solve the quadratic equation $$2x^2 + 8x + 12 = 0.$$\n\n2. Start by finding the discriminant $$\Delta$$ which tells us the nature of the roots: $$\Delta = b^2

1. Solve $x^2 - 2x - 15 = 0$ using quadratic formula: $$a=1, b=-2, c=-15$$

1. To provide a detailed explanation without using sigma notation, I will break down the process into clear, numbered steps. 2. Sigma notation ($\sum$) is a compact way of represen

1. **State the problem:** Prove that the sum of binomial coefficients from 0 to n equals $2^n$, i.e., $$\sum_{r=0}^n {n \choose r} = 2^n.$$

1. **Problem Statement:** Consider the following scenarios:

1. **State the problem:** Solve the simultaneous equations: $$2\log y = \log 2 + \log x$$