🧮 algebra

1. **State the problem:** Find the product and simplify the expression $$-4t^{2}(4t^{2} - t + 1)$$. 2. **Formula and rules:** To multiply a monomial by a polynomial, multiply the m

1. **State the problem:** Find the product and simplify the expression $$(4w + 3)(-3w^2 - 3w - 3).$$ 2. **Recall the distributive property:** To multiply two polynomials, multiply

1. **State the problem:** Evaluate the expression $2r + 3(5k - 6)$ when $x = 2$. 2. **Clarify variables:** The expression contains variables $r$ and $k$, but the problem gives a va

1. Problema este să calculăm produsul a două fracții: $\frac{26}{3} \times \frac{27}{13}$.\n\n2. Formula pentru înmulțirea fracțiilor este: $$\frac{a}{b} \times \frac{c}{d} = \frac

1. **State the problem:** Simplify the expression $-5c - 4c + ab + 10$ given $c=2$ and $d=4$. Note that $d$ is not present in the expression, so it will not affect the simplificati

1. **Problem c)**: Given the proportion $\frac{7}{9} : \frac{4}{7} = \frac{m}{n} : 729$, find $m$ and $n$. 2. **Step 1:** Understand the proportion. The ratio $\frac{7}{9}$ to $\fr

1. **State the problem:** Given the equation $$(a^b)^n = \frac{256}{729}$$, find the values of $a$, $b$, and $n$. 2. **Rewrite the equation using exponent rules:** Recall that $$(a

1. **State the problem:** Solve the equation $$|3x| = 24$$ for $x$. 2. **Recall the absolute value property:** For any real number $a$, $$|a| = b$$ implies $$a = b$$ or $$a = -b$$.

1. **Problem Statement:** Show that if $\alpha$ and $\beta$ are roots of the quadratic equation $$x^2 - 14x + 36 = 0,$$ then for all positive integers $n$, the expression $$\alpha^

1. **State the problem:** We are given points on an exponential function: $(-1, \frac{1}{7})$, $(0, 1)$, $(1, 7)$, and $(2, 49)$. We want to show that the function grows by equal f

1. **Problem Statement:** Given a parabola that opens upward, crosses the x-axis near -5 and -3, has a vertex near (-4, -9), and does not cross the y-axis.

1. **State the problem:** Solve the equation $ (3x - 8)(4x - 5) = 0 $ for $x$. 2. **Formula and rule:** The zero product property states that if a product of two factors equals zer

1. **State the problem:** Solve the equation $ (3x - 8)(4x - 5) = 0 $ for $x$. 2. **Formula and rule:** If a product of two factors equals zero, then at least one of the factors mu

1. Problemet: Vi skal finde den afledte af funktionen $$f(x) = \ln(x) \cdot (5x^4 + 2)$$ og derefter bestemme $$f'(1)$$. 2. Formel: Vi bruger produktreglen for differentiation, som

1. The problem asks to find $f(g(3))$ given the functions $f(x) = 3x - 7$ and the graph of $g$. 2. From the graph, find the value of $g(3)$. Observing the graph, at $x=3$, $g(3) =

1. **Enunciare il problema:** Discutere e risolvere il sistema lineare a parametri reali $k$:

1. **State the problem:** We want to find an approximate solution to the equation $$x^3 - 4x + 1 = 0$$ using the iterative formula $$x_{n+1} = - \frac{1}{x_n^2 - 4}$$ starting with

1. **State the problem:** We want to find an approximate solution to the equation $$x^3 - 5x^2 - 12 = 0$$ using the iterative formula $$x_{n+1} = 5 + \frac{12}{x_n^2}$$ starting wi

1. **State the problem:** We need to find the discount percentage when the original price of a couch is 405 and the discounted price is 162.

1. **State the problem:** A store owner discounted some rare acoustic guitars from $7,325 to $2,344. We need to find the discount percentage. 2. **Formula for discount percentage:*

1. **Problem Statement:** We need to write an equation for a polynomial function given its graph.