Polynomials
The height of a drone as a function of time can be written as $h(t) = -2t^2 + 12t + c$: with a single expression you know its height at any moment. That is a polynomial. You already worked with them in 2nd and 3rd ESO; here you review their parts with slightly more general examples.
Degree, leading coefficient and constant term
$$P(x) = -5x^4 + \frac{2}{3}x^2 - x + 9$$-
Degree of the polynomial: the highest degree among its terms (here, $4$).
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Leading coefficient: the coefficient of the term of highest degree ($-5$).
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Constant term: the term that does not contain $x$ ($9$).
Ordering and completing a polynomial
A polynomial is complete if it has all the degrees from the highest down to the constant term. If any is missing, it is incomplete.
$x^3 - 4$ is incomplete: it is missing the terms of degree $2$ and degree $1$. To divide or subtract in columns, it is useful to leave a gap (or write a $0$) wherever a degree is missing.
Numerical value of a polynomial
The numerical value of $P$ at a point is obtained by substituting each letter with its value (in brackets if it is negative) and calculating, following the order of operations.
Calculate the numerical value of the polynomial $P(x, y) = 3x^2y - 5xy^2 + 2x - 3y + 1$ for $x = -2$ and $y = 3$.
Calculate the numerical value of the polynomial $P(x, y) = 3x^2y - 5xy^2 + 2x - 3y + 1$ for $x = -2$ and $y = 3$.
Roots: a first idea
Using the factor theorem, check whether $x = -3$ is a root of the polynomial $P(x) = x^3 + 4x^2 + x - 6$.
Using the factor theorem, check whether $x = -3$ is a root of the polynomial $P(x) = x^3 + 4x^2 + x - 6$.
Polynomials in real-life problems
Prices, areas or trajectories are often described by a polynomial; the question is answered by calculating its numerical value (or solving for an unknown) in the situation of interest.
The height $h$ (in metres) of a drone as a function of time $t$ (in seconds) is given by the polynomial $h(t) = -2t^2 + 12t + c$. If at $3$ seconds the drone is at a…
The height $h$ (in metres) of a drone as a function of time $t$ (in seconds) is given by the polynomial $h(t) = -2t^2 + 12t + c$. If at $3$ seconds the drone is at a height of $25$ metres, calculate the value of $c$ and determine the initial height from which it took off.
A technology company estimates that its monthly revenue (in thousands of euros) is given by the polynomial $I(x) = -2x^2 + 50x$, and its production costs by…
A technology company estimates that its monthly revenue (in thousands of euros) is given by the polynomial $I(x) = -2x^2 + 50x$, and its production costs by $C(x) = 10x + 150$, where $x$ is the quantity of devices manufactured (in hundreds). Obtain the polynomial representing the monthly profit $B(x)$ and calculate the exact profit if $1000$ devices are manufactured.
Back to the start: If the drone takes off from the ground, its initial height is $h(0) = c = 0\text{ m}$; if it takes off from a table at $1{,}2\text{ m}$, the constant term is $c = 1{,}2\text{ m}$ and it determines the take-off height.