🧮 algebra

1. **State the problem:** Solve the inequality $4^{x+3} \geq 4^x$. 2. **Recall the properties of exponents:** For any base $a > 0$ and $a \neq 1$, if $a^m \geq a^n$, then $m \geq n

1. The problem is to find the value of $a$ such that the line $y = -1.5$ intersects the parabola $y = x^2 + 8x + a$. 2. Set the two expressions for $y$ equal to each other to find

1. **State the problem:** We have 5 points A(-3,4), B(-3,8), C(0,12), D(3,8), and E(3,8) on a petal and want to find the equation of the line passing through these points. 2. **Che

1. **State the problem:** We are given the function $g(x) = x - x^3 + 1$ and want to analyze it. 2. **Write the function:**

1. The problem asks us to form a polynomial with given real zeros $-2$, $-1$, $2$, and $3$, and degree $4$. 2. The polynomial can be expressed as the product of factors correspondi

1. **Problem statement:** Form a polynomial with zeros 6 (multiplicity 1) and 3 (multiplicity 2), and degree 3. 2. **Recall:** A polynomial with zeros $r$ of multiplicity $m$ inclu

1. **State the problem:** Find a cubic polynomial function $f(x)$ with zeros at $-2$, $3$, and $5$, and that passes through the point $(7,144)$. 2. **Write the general form:** Sinc

1. **State the problem:** Find a polynomial function $f(x)$ of degree 4 with real zeros $-6, 0, 3, 2$ and that passes through the point $\left(-\frac{1}{2}, -385\right)$. 2. **Writ

1. **State the problem:** Find a polynomial function of degree 4 with zeros at $-2$ and $2$, each with multiplicity 2, and whose graph passes through the point $(-4, 432)$. 2. **Wr

1. **State the problem:** Construct a polynomial function $f(x)$ of degree 4 with zeros at $-1$ (multiplicity 1), $2$ (multiplicity 2), and $3$ (multiplicity 1), and that passes th

1. **Problem Statement:** Given the polynomial function $$f(x) = 6 (x^2 + 4)^2 (x - 3)^3,$$ answer the following: (a) List each real zero and its multiplicity.

1. **State the problem:** Given the polynomial function $$f(x) = -10 \left(x + \frac{4}{5}\right)^2 (x + 4)^3,$$ we need to find: (a) The real zeros and their multiplicities.

1. **Problem Statement:** Given the polynomial function $$f(x) = (x - 7)^3 (x + 8)^2,$$ we need to find: (a) The real zeros and their multiplicities.

1. **State the problem:** We have the polynomial function $$f(x) = \frac{1}{3} (3x^2 + 8)^2 (x^2 + 9)$$ and need to find its real zeros and their multiplicities, determine if the g

1. **State the problem:** We are given the polynomial function $$f(x) = -7x^2(x^2 - 3)$$ and need to find: (a) The real zeros and their multiplicities.

1. **Problem Statement:** Determine if the given graph could represent a polynomial function. If yes, list the real zeros and state the least degree the polynomial can have. 2. **U

1. **State the problem:** We are given the function $$G(x) = (x + 1)^2 (x - 1)$$ and asked to find:

1. **State the problem:** Find the real zeros of the function $$f(x) = 6(x^2 - 9)(x^2 + 12x + 27)^2$$ and determine their multiplicities. 2. **Recall the formula and rules:** To fi

1. The problem is to identify which quadratic equation best matches the given graph. 2. The general form of a quadratic equation is $$y = ax^2 + bx + c$$ where $a$ determines the c

1. **State the problem:** We need to determine which quadratic equation best matches the given graph. 2. **Recall the general form of a quadratic equation:**

1. The problem is to simplify the expression $$\sqrt[3]{32}$$ and evaluate $$\left(\frac{125}{1000}\right)^{\frac{1}{3}}$$. 2. Recall that the cube root of a number $a$ is the numb