Polynomial Basics
What is the degree of 3x⁴ + 2x² - x + 7?
4 (Degree is the highest exponent: 3x⁴ → degree is 4)
Simplify: (2x² + 3x - 4) + (x² - 5x + 6).
3x² - 2x + 2 (Combine like terms)
Площадь правильного треугольника равна 3x2−5x+2, а высота равна x−1. Найдите периметр этого треугольника.
18x −12
The volume of a cube is x³. Write an expression for its surface area.
6x² (Surface Area of a Cube)
I used to be (x + 2)(x - 2),
but now I’m stuck in therapy, wondering who I really am. Who am I after expanding?
x² - 4 — a classic difference of squares having an existential moment!
Найдите многочлен, у которого равны между собой старший коэффициент, свободный член, количество членов и степень многочлена.
2x2 +2
Expand: (x - 3)(x + 5).
x² + 2x - 15 (Use FOIL method)
The height of a triangle is x + 2, and its base is 2x - 1. Write an expression for the area.
½ (x + 2)(2x - 1) = (2x² + 3x - 2) / 2
A farmer fences a rectangular field with a length of 2x + 3 and a width of x + 5. Find the total length of the fence.
6x + 16 (Perimeter = 2L+2W2L + 2W2L+2W)
I’m a friendly little polynomial pet.
I always return to my owner — no matter what value of x you plug in, my expression comes out the same!
What kind of polynomial am I?
A constant polynomial — loyal like a dog, always giving you the same value!
Write a trinomial with degree 3 and a constant term of 7.
Example: x³ + 2x² + 7
Разложите на множители x2 + y2 +z2 −2xy −2yz +2xz
(x −y+z)·(x−y+z)
The perimeter of a triangle is 3x² - x + 5. Two of its sides are x² - 2x + 3 and x² + x - 1. Find the third side.
x² - 2x + 3 + x² + x - 1 = 2x² - x + 2 → Third side: x² + 3
A box has a length of x + 3, width x - 2, and height x + 1. Write an expression for its volume.
(x + 3)(x - 2)(x + 1) = x³ + 2x² -5x - 6
I’m hiding inside every expression, but no one sees me. You can add me, subtract me —
and I still leave everything unchanged.
Who am I?
Zero!
The ultimate undercover agent in math — always there, but never messing things up. 😎
How many terms are in the expression x³ + 2x² - 5x + 4x² - 3 after simplifying?
4 terms (After simplifying: x³ + 6x² - 5x - 3 terms)
Factor completely: x³ - 3x² - 4x + 12.
(x - 3)(x - 2)(x + 2) (Factor step by step)
A square’s area is x² + 8x + 16. What is its side length?
(x + 4)² = x² + 8x + 16 → Side length: x + 4
A trapezoid has bases (x + 4) and (x - 2) and height x + 1. Write an expression for its area.
(½) (x + 4 + x - 2)(x + 1) = ½ (2x + 2)(x + 1) = (x² + 2x + 1)
В русскоязычной литературе для преобразования многочленов в стан дартный вид используется конкретный глагол. Однако он же в совершенном виде применяется далеко не к каждому многочлену стандартного вида. Назовите старший коэффициент в многочлене, к кото рому применён рассматриваемый глагол в совершенном виде
1
Find a polynomial with exactly 3 terms, a leading coefficient of 4, and a constant of -6.
Example: 4x³ + 2x - 6
Divide: (x³ + 3x² - x - 3) ÷ (x + 1) using long division.
x² + 2x - 3 (Using polynomial long division)
A park is (x² + 5x + 6) m² in area. A new path increases the area to (x² + 7x + 12) m². Find the path’s area.
(x² + 7x + 12) - (x² + 5x + 6) = 2x + 6
Объём цилиндрического стержня равен xy − x + y − 1, а площадь всей его поверхности равна x + 1. Считая, что стержень является фигурой вращения прямоугольника вокруг своей стороны, найдите отношение площади этого прямоугольника к его периметру.
y −1
You call me by many names — binomial, trinomial, quadratic, cubic…But deep down, we’re all just expressions trying to do one thing.
What is it?
Solve for x — we all just want to find our roots!/ be “Equal zero.”