Operations with polynomials
In a strategy video game you have two chests. The first contains $P(x) = 5x^2 + 3x - 4$ items (depending on their rarity $x$) and the second contains $Q(x) = 2x^2 - 5x + 8$. To find out how many you have in total, you add the two polynomials, term by term.
Adding and subtracting polynomials
You add (or subtract) the like terms, that is, the terms of the same degree. It helps to line them up in columns, just like in primary-school addition:
In a strategy video game, your inventory is divided into two chests. The contents of the first chest are given by the polynomial $P(x) = 5x^2 + 3x - 4$ and that of the…
In a strategy video game, your inventory is divided into two chests. The contents of the first chest are given by the polynomial $P(x) = 5x^2 + 3x - 4$ and that of the second by $Q(x) = 2x^2 - 5x + 8$, where $x$ represents the rarity level of the items. Calculate the total polynomial representing the sum of the two chests: $P(x) + Q(x)$.
To organize an esports event, a company uses the expression $G(x) = 10x^2 - 4x + 100$ to calculate its expenses and $I(x) = 15x^2 + 2x$ to calculate its income (where…
To organize an esports event, a company uses the expression $G(x) = 10x^2 - 4x + 100$ to calculate its expenses and $I(x) = 15x^2 + 2x$ to calculate its income (where $x$ is the number of sponsors). Find the polynomial that represents the net profit, knowing that profit is the difference between income and expenses: $I(x) - G(x)$.
Monomial times polynomial
You apply the distributive property: the monomial multiplies each term of the polynomial.
$$3x^2 \cdot (2x^3 - 4x + 5) = 6x^5 - 12x^3 + 15x^2$$Perform the following operation by multiplying the monomial by the polynomial (applying the distributive property): $3x^2 \cdot (2x^3 - 4x + 5)$
Perform the following operation by multiplying the monomial by the polynomial (applying the distributive property):
$3x^2 \cdot (2x^3 - 4x + 5)$
Polynomial times polynomial
Each term of the first polynomial multiplies each term of the second. A rectangle shows this very well:
A young designer is creating a logo for his clothing brand. The central logo is a rectangle like the one in the figure. Find the polynomial that represents the total…
A young designer is creating a logo for his clothing brand. The central logo is a rectangle like the one in the figure. Find the polynomial that represents the total area of the figure.
Taking out a common factor
This is the distributive property in reverse: if all the terms have something in common, you take it out and multiply it by a bracket.
$$4x^2 + 8x = 4x\,(x + 2)$$Factorise out the common factor in the following simple algebraic expressions: a) $4x^2 + 8x$ b) $5a^3 - 10a^2 + 15a$
Factorise out the common factor in the following simple algebraic expressions:
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a) $4x^2 + 8x$
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b) $5a^3 - 10a^2 + 15a$
Elements and numerical value: a review
Before working with a polynomial, always identify its degree, its leading coefficient, its constant term and how many terms it has.
Identify the main elements of the following polynomial: $P(x) = -7x^4 + 2x^3 - x + 12$ a) What is its degree? b) What is the leading coefficient? c) What is the constant…
Identify the main elements of the following polynomial:
$P(x) = -7x^4 + 2x^3 - x + 12$
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a) What is its degree?
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b) What is the leading coefficient?
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c) What is the constant term?
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d) How many terms does it have?
Evaluate the numerical value of the polynomial $A(x) = 3x^3 - 5x^2 + 2x - 7$ for the following cases: a) For $x = 0$ b) For $x = 1$
Evaluate the numerical value of the polynomial $A(x) = 3x^3 - 5x^2 + 2x - 7$ for the following cases:
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a) For $x = 0$
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b) For $x = 1$
Back to the beginning: Adding the items in both chests gives $P(x) + Q(x) = (5x^2 + 3x - 4) + (2x^2 - 5x + 8) = 7x^2 - 2x + 4$ items in total.