Polynomials. Factorisation
A polynomial such as $B(x)=-x^3+15x^2-44x+30$ can describe the weekly profit (in thousands of euro) of a Galway start-up selling vintage clothing online as a function of how much (in thousands of euro) is invested in marketing. Factorising it completely, which means 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: expanding and factorising expressions. If you want to go further, there are also optional parts on dividing polynomials and finding their roots.
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.
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) $(x + 7)(x - 7)$
Polynomial division
Dividing the dividend $D(x)$ by the divisor $d(x)$ 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 $x-a$, there is a shortcut that works only with the coefficients.
Apply Ruffini's rule to factorise the polynomial $P(x) = x^3 - 6x^2 + 11x - 6$, knowing that $x = 1$ is one of its roots. Write the polynomial as a product of three…
Apply Ruffini's rule to factorise the polynomial $P(x) = x^3 - 6x^2 + 11x - 6$, knowing that $x = 1$ is one of its roots. Write the polynomial as a product of three linear factors.
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.
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 $P(a)=0$.
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.
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.
Write the expression for a monic polynomial (one whose leading coefficient is 1) of degree three knowing that its roots are $x = 0$, $x = 2$ and $x = -3$. Express the…
Write the expression for a monic polynomial (one whose leading coefficient is 1) of degree three knowing that its roots are $x = 0$, $x = 2$ and $x = -3$. Express the final result as a product of factors.
Factorising: common factor, perfect squares and quadratics
Factorising always starts the same way: take out the highest common factor (HCF). Then look for a difference of two squares, a perfect square or a quadratic trinomial.
Find the highest common factor (HCF) and the lowest common multiple (LCM) of the following polynomials: $P(x) = x^3 - 4x$ $Q(x) = x^3 - 4x^2 + 4x$
Find the highest common factor (HCF) and the lowest common multiple (LCM) of the following polynomials:
$P(x) = x^3 - 4x$
$Q(x) = x^3 - 4x^2 + 4x$
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:
$I(x) = 12x^5 - 18x^4 + 6x^2$
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 $x$.
A rectangular display screen is being designed whose surface area in square metres is described by the polynomial $S(x) = x^2 - 16$. a) Factorise the polynomial using…
A rectangular display screen is being designed whose surface area in square metres is described by the polynomial $S(x) = x^2 - 16$.
- a) Factorise the polynomial using the difference of two squares identity.
- b) If $x$ must be greater than 4 for the dimensions to make sense, what would be the expressions for the base and height of the screen?
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.
Apply the Rational Root Theorem to find all real roots of the polynomial $P(x) = 2x^3 + 3x^2 - 8x + 3$. Once found, write its factorisation.
Apply the Rational Root Theorem to find all real roots of the polynomial $P(x) = 2x^3 + 3x^2 - 8x + 3$. Once found, write its factorisation.
In a study on population biology, the growth of a bacterial colony has been modelled using the degree six polynomial $B(t) = t^6 - 9t^3 + 8$, where $t$ is the time in…
In a study on population biology, the growth of a bacterial colony has been modelled using the degree six polynomial $B(t) = t^6 - 9t^3 + 8$, where $t$ is the time in hours. Biologists need to know at which exact moments the population is zero to understand the extinction cycles.
Factorise the polynomial $B(t)$ using a suitable change of variable to convert it into a quadratic polynomial. Express the result as a product of linear factors and irreducible quadratic factors over the set of real numbers.
Back to the beginning: For $B(x)=-x^3+15x^2-44x+30$, (beyond the syllabus for this level) the profit is zero at $x=1$, since $B(1)=0$, and dividing by $(x-1)$ by long division 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 euro, that is €1,000, about €2,640 and about €11,360 respectively.