Polynomials. Factorisation
A polynomial such as $B(x)=-x^3+15x^2-44x+30$ can describe the weekly profit of a startup as a function of how much is invested in marketing. Factorising it completely —finding its roots and writing it as a product of simple factors— lets us answer questions like "for which investments is the profit zero?" without having to try values blindly. This section brings together everything you need: operating with polynomials, dividing them, and factorising them all the way.
Definition, degree and basic operations
Adding and subtracting polynomials means collecting like terms (terms of the same degree). Multiplying means applying the distributive property: each term of the first polynomial times each term of the second, and then simplifying.
In designing a new level for a video game, the energy of a protective shield follows the polynomial function $P(x) = x^3 - 5x^2 + 7x - 3$, where $x$ represents the time…
In designing a new level for a video game, the energy of a protective shield follows the polynomial function $P(x) = x^3 - 5x^2 + 7x - 3$, where $x$ represents the time in seconds since activation. Use the Remainder Theorem to check whether the shield runs out of energy at exactly $x = 3$ seconds.
Simplify the following expression by taking out the highest common factor from the terms of the polynomial that represents the revenue of a small custom t-shirt…
Simplify the following expression by taking out the highest common factor from the terms of the polynomial that represents the revenue of a small custom t-shirt business:
$I(x) = 12x^5 - 18x^4 + 6x^2$
Polynomial division and Ruffini's rule
Dividing $D(x)$ by $d(x)$ gives a quotient $C(x)$ and a remainder $R(x)$ with $D(x)=d(x)\cdot C(x)+R(x)$, always with the remainder of lower degree than the divisor. If the divisor has the form $x-a$, the shortcut is Ruffini's rule: you work only with the coefficients.
Expand the following expressions using algebraic identities. These expressions often appear when calculating variations in area on architectural plans: a) $(2x + 5)^2$…
Expand the following expressions using algebraic identities. These expressions often appear when calculating variations in area on architectural plans:
- a) $(2x + 5)^2$
- b) $(3x - 2)^2$
- c) $(x + 7)(x - 7)$
Factorise the following polynomials using algebraic identities and taking out common factors: a) $R(x) = 4x^3 - 12x^2 + 9x$ b) $S(x) = x^4 - 81$ c)…
Factorise the following polynomials using algebraic identities and taking out common factors:
- a) $R(x) = 4x^3 - 12x^2 + 9x$
- b) $S(x) = x^4 - 81$
- c) $T(x) = 3x^2 + 30x + 75$
The remainder theorem and the factor theorem
Before dividing, it is worth knowing whether it is necessary: the remainder theorem says that the remainder of dividing $P(x)$ by $(x-a)$ is, directly, $P(a)$. The factor theorem says that $(x-a)$ is a factor of $P(x)$ exactly when $P(a)=0$.
A farmer has a square plot and has decided to enlarge it. The area of the new plot, which is also square, is given by the trinomial $A(x) = x^2 + 10x + 25$, where $x$ is…
A farmer has a square plot and has decided to enlarge it. The area of the new plot, which is also square, is given by the trinomial $A(x) = x^2 + 10x + 25$, where $x$ is a length in meters and the area is in square meters. Factor this expression using special identities (square of a sum) to find the length of the side of the new plot in terms of $x$.
Consider an unknown polynomial $P(x)$ for which we only have partial information obtained through computer lab tests. We know that when dividing $P(x)$ by $(x - 3)$, the…
Consider an unknown polynomial $P(x)$ for which we only have partial information obtained through computer lab tests. We know that when dividing $P(x)$ by $(x - 3)$, the remainder obtained is $10$. On the other hand, when dividing the same polynomial $P(x)$ by $(x + 2)$, the remainder is $-5$.
Determine, giving reasons, what the remainder of dividing the polynomial $P(x)$ by the product $(x - 3)(x + 2)$ will be. Hint: Remember that the degree of the remainder is always less than the degree of the divisor.
Factorisation: common factor and special identities
Factorising always starts the same way: take out as much common factor as possible and, if what remains is a special identity or a quadratic equation, solve it.
Determine the value of the unknown coefficient $k$ so that the polynomial $P(x) = x^3 - 2x^2 + kx - 10$ is divisible by $(x - 5)$. Remember that to be divisible, the…
Determine the value of the unknown coefficient $k$ so that the polynomial $P(x) = x^3 - 2x^2 + kx - 10$ is divisible by $(x - 5)$. Remember that to be divisible, the remainder of the division must be zero.
Completely factorise the following polynomial by decomposing it into the product of as many linear factors as possible: $P(x) = 2x^4 - 4x^3 - 10x^2 + 12x$ To do so,…
Completely factorise the following polynomial by decomposing it into the product of as many linear factors as possible:
$P(x) = 2x^4 - 4x^3 - 10x^2 + 12x$
To do so, factorise by taking out the common factor, use Ruffini's rule and, if necessary, solve the resulting quadratic equation.
Complete factorisation: rational roots and irreducible factors
When the leading coefficient is not $1$, as well as the divisors of the constant term we have to try fractions: candidates $\frac{p}{q}$, with $p$ a divisor of the constant term and $q$ a divisor of the leading coefficient. And a factor of degree $2$ with a negative discriminant is irreducible: that is where the factorisation stops.
Calculate the value of the parameter $k$ so that the polynomial $P(x) = x^3 - kx^2 + 5x + 12$ is divisible by the binomial $(x + 1)$. Once the value of $k$ is found,…
Calculate the value of the parameter $k$ so that the polynomial $P(x) = x^3 - kx^2 + 5x + 12$ is divisible by the binomial $(x + 1)$. Once the value of $k$ is found, find the remaining roots of the polynomial.
Factorise the polynomial $P(x) = 12x^3 + 8x^2 - 13x + 3$ knowing that it has no integer roots. You must apply the Rational Root Theorem to identify the possible…
Factorise the polynomial $P(x) = 12x^3 + 8x^2 - 13x + 3$ knowing that it has no integer roots. You must apply the Rational Root Theorem to identify the possible candidates of the form $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.
Once the first rational root has been found, use Ruffini's rule and solve the resulting quadratic equation to find the other roots and write the complete factorisation.
Back to the beginning: For $B(x)=-x^3+15x^2-44x+30$, by trying divisors we find the root $x=1$, and dividing using Ruffini's rule leaves $-(x-1)(x^2-14x+30)$; solving the quadratic equation, the profit is zero for investments of $x=1$, $x=7-\sqrt{19}\approx 2{,}64$ and $x=7+\sqrt{19}\approx 11{,}36$ thousand euros.