Operations with polynomials
If you are designing the HUD of a video game and each object is worth $P(x) = 3x^2 - 2x + 5$ coins depending on its level $x$, working out how many coins a batch of $Q(x)$ objects gives is a matter of multiplying two polynomials. They are the same rules you already know, only now it pays to work quickly.
Adding and subtracting polynomials
You add or subtract the like terms, the ones of the same degree. Line them up in columns:
Given the polynomials $P(x) = 3x^3 - 2x^2 + 5x - 1$ and $Q(x) = -x^3 + 4x^2 - 5x + 3$, calculate the expressions resulting from the operations $P(x) + Q(x)$ and…
Given the polynomials $P(x) = 3x^3 - 2x^2 + 5x - 1$ and $Q(x) = -x^3 + 4x^2 - 5x + 3$, calculate the expressions resulting from the operations $P(x) + Q(x)$ and $P(x) - Q(x)$.
Add the polynomials $A(x) = x^2 - 3x + 2$ and $B(x) = 2x^2 + x - 5$. Once you have obtained the resulting polynomial, calculate its numerical value for $x = -2$.
Add the polynomials $A(x) = x^2 - 3x + 2$ and $B(x) = 2x^2 + x - 5$. Once you have obtained the resulting polynomial, calculate its numerical value for $x = -2$.
Monomial times polynomial
Distributive property: the monomial multiplies every term.
$$-4x^2 \cdot (2x^3 - \tfrac{1}{2}x^2 + 3x - 5) = -8x^5 + 2x^4 - 12x^3 + 20x^2$$Perform the multiplication of the monomial $-4x^2$ by the polynomial $2x^3 - \frac{1}{2}x^2 + 3x - 5$ and write the result in descending order of degree.
Perform the multiplication of the monomial $-4x^2$ by the polynomial $2x^3 - \frac{1}{2}x^2 + 3x - 5$ and write the result in descending order of degree.
Polynomial times polynomial
Every term of the first polynomial multiplies every term of the second. A rectangle of areas shows it clearly:
Calculate the product of the following polynomials and collect like terms: $(2x - 3) \cdot (x^2 + 4x - 1)$.
Calculate the product of the following polynomials and collect like terms: $(2x - 3) \cdot (x^2 + 4x - 1)$.
A rectangular garden has dimensions that depend on a value $x$. Its length is $3x + 2$ metres and its width is $x + 5$ metres. Express the total area of the garden as a…
A rectangular garden has dimensions that depend on a value $x$. Its length is $3x + 2$ metres and its width is $x + 5$ metres. Express the total area of the garden as a polynomial in $x$ and calculate how many square metres it will have if $x = 4$.
Special products (notable identities): a quick revision
They are such common products that it is worth recognising them without expanding step by step. Revise them with the interactive if you are a bit rusty.
Expand the following expression using the standard identity formula for the square of a sum: $(3x + 5y)^2$.
Expand the following expression using the standard identity formula for the square of a sum: $(3x + 5y)^2$.
Calculate the result of the following operation using the product of a sum and a difference identity: $(7a + 2b) \cdot (7a - 2b)$.
Calculate the result of the following operation using the product of a sum and a difference identity: $(7a + 2b) \cdot (7a - 2b)$.
Expand and simplify fully the following algebraic expression by applying the corresponding standard identities: $(2x + 3)^2 - (2x - 3)^2 + (x - 1) \cdot (x + 1)$
Expand and simplify fully the following algebraic expression by applying the corresponding standard identities:
$(2x + 3)^2 - (2x - 3)^2 + (x - 1) \cdot (x + 1)$
Degree and leading coefficient of a product
Before multiplying everything out, sometimes you only need to know the degree and the coefficient of the highest-degree term of the result: you get them by multiplying the leading coefficients and adding the degrees, without expanding everything else.
When multiplying the polynomial $A(x) = 5x^4 - 3x + 2$ by the polynomial $B(x) = -2x^3 + x^2$, state the degree of the resulting polynomial and the value of its leading…
When multiplying the polynomial $A(x) = 5x^4 - 3x + 2$ by the polynomial $B(x) = -2x^3 + x^2$, state the degree of the resulting polynomial and the value of its leading coefficient, without performing the full multiplication.
Given the polynomials $A(x) = x^3 - 4x^2 + x - 1$ and $B(x) = 2x^3 + x^2 - 5$, determine the expression for an unknown polynomial $P(x)$ that satisfies the following…
Given the polynomials $A(x) = x^3 - 4x^2 + x - 1$ and $B(x) = 2x^3 + x^2 - 5$, determine the expression for an unknown polynomial $P(x)$ that satisfies the following equality:
$2 \cdot A(x) - P(x) = B(x)$
Back to the start: If you have a batch of $Q(x) = 2x + 1$ objects, the total loot obtained is $(3x^2 - 2x + 5)(2x + 1) = 6x^3 - x^2 + 8x + 5$ coins.