Polynomials
A water rocket is launched from a platform. Its height in metres, after a time $t$ in seconds, is $h(t) = -5t^2 + 25t + 2$. With a single formula you know the height at any moment: that is a polynomial in action.
What a polynomial is
$$P(x) = -5x^4 + \frac{2}{3}x^2 - x + 9$$-
Degree of the polynomial: the highest degree of 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 has no $x$ ($9$).
Analyse the following polynomial and answer the questions: $P(x) = -2x^3 + 5x^2 - 8$ a) What is the degree of the polynomial? b) What is the name of the term that has no…
Analyse the following polynomial and answer the questions:
$P(x) = -2x^3 + 5x^2 - 8$
- a) What is the degree of the polynomial?
- b) What is the name of the term that has no variable ($-8$)?
- c) How many terms does it have? Based on the number of terms, is it a monomial, binomial or trinomial?
- d) What is the coefficient of the highest degree term?
Putting a polynomial in order and completing it
A polynomial is complete if it has every degree 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 work in columns, it helps to leave a gap (or a $0$) where a degree is missing.
Copy and complete the following table in your exercise book to identify the fundamental elements of these monomials: | Monomial | Coefficient | Literal Part | Degree | |…
Copy and complete the following table in your exercise book to identify the fundamental elements of these monomials:
| Monomial | Coefficient | Literal Part | Degree |
|---|---|---|---|
| $-7x^4$ | |||
| $\frac{2}{3}ab^2$ | |||
| $xyz$ | |||
| $5$ |
Numerical value of a polynomial
The numerical value of $P(x)$ at $x = a$ is written $P(a)$: replace $x$ by $a$ (in brackets) and calculate.
A pizza delivery driver earns a fixed wage of $15$ euros for each evening of work, plus an extra $3$ euros for each home delivery order. The expression that calculates…
A pizza delivery driver earns a fixed wage of $15$ euros for each evening of work, plus an extra $3$ euros for each home delivery order. The expression that calculates their earnings is $G = 15 + 3x$, where $x$ is the number of orders.
Calculate the numerical value of their earnings (the total amount they will receive) if one evening they deliver $8$ orders.
Roots of a polynomial
Given the following polynomial: $P(x) = -5x^4 + \frac{2}{3}x^2 - x + 9$ State with reasoning the following elements: a) The degree of the polynomial. b) The coefficient…
Given the following polynomial:
$P(x) = -5x^4 + \frac{2}{3}x^2 - x + 9$
State with reasoning the following elements:
- a) The degree of the polynomial.
- b) The coefficient of the degree 2 term.
- c) The constant term.
- d) The numerical value of $P(x)$ for $x = -1$.
Polynomials in real-life problems
Many situations (prices, areas, trajectories) are described by a polynomial and are solved by calculating its numerical value at the moment of interest.
A small water rocket is launched from a platform. The height $h$ (in metres) of the rocket as a function of time $t$ (in seconds) is given by the polynomial:…
A small water rocket is launched from a platform. The height $h$ (in metres) of the rocket as a function of time $t$ (in seconds) is given by the polynomial:
$h(t) = -5t^2 + 25t + 2$
- a) At what height is the launch platform? (Hint: calculate $h(0)$).
- b) At what height will the rocket be 2 seconds after being launched?
- c) Will the rocket be higher at 3 seconds or at 4 seconds? Justify your answer by calculating the corresponding numerical values.
In the world of video games, the scoring system of a new platform title works as follows: for each gold coin collected you receive an unknown amount of points $x$, but…
In the world of video games, the scoring system of a new platform title works as follows: for each gold coin collected you receive an unknown amount of points $x$, but for each enemy defeated you receive three times as many points as for a coin. At the end of the level, you always get a fixed bonus of $50$ experience points.
Write the algebraic expression representing the total score of a player who has collected $12$ coins and defeated $4$ enemies.