Factoring Polynomials
A box manufacturer in Puebla tells you that the volume of its box is $V(x)=x^3-7x^2+10x$. How long, wide, and tall is it? Factoring a polynomial means "taking it apart" into its pieces: $P(x)=2x(x+5)(x-4)^2$ tells you at a glance that $P(x)=0$ only when $x=0$, $x=-5$, or $x=4$. Here we put all the techniques together into one procedure, and at the end we solve the box problem.
Common factor and special products
Factoring always starts the same way: pull out all the common factor you can. After that, if what is left is a special product (difference of squares, perfect square trinomial) or a second-degree trinomial, you factor it with those or with the quadratic formula.
Factorise out the common factor and use algebraic identities to fully factorise the following polynomial: $P(x) = 3x^5 - 12x^3$
Factorise out the common factor and use algebraic identities to fully factorise the following polynomial:
$P(x) = 3x^5 - 12x^3$
Factorise the polynomial $P(x) = 2x^2 + 5x - 3$ by first solving the associated quadratic equation.
Factorise the polynomial $P(x) = 2x^2 + 5x - 3$ by first solving the associated quadratic equation.
Factorise the polynomial $P(x) = (x^2 - 4)^2 - (x + 2)^2$ using algebraic identities and factorising out common factors, without needing to expand the entire polynomial…
Factorise the polynomial $P(x) = (x^2 - 4)^2 - (x + 2)^2$ using algebraic identities and factorising out common factors, without needing to expand the entire polynomial first.
Roots, the remainder theorem, and synthetic division
If $r$ is a root of $P(x)$, then $(x-r)$ is a factor. To check this, you just evaluate: by the remainder theorem, the remainder of dividing $P(x)$ by $(x-r)$ is $P(r)$. If $P(r)=0$, the division is exact. To do the division quickly you use synthetic division (Ruffini's rule).
The candidate integer roots are the divisors of the constant term.
Check, using the Remainder Theorem, whether $x = -1$ is a root of the polynomial $P(x) = x^4 + 3x^3 - 2x^2 + x + 7$. If so, factorise out the corresponding factor.
Check, using the Remainder Theorem, whether $x = -1$ is a root of the polynomial $P(x) = x^4 + 3x^3 - 2x^2 + x + 7$. If so, factorise out the corresponding factor.
Factorise the following cubic polynomial knowing that one of its roots is $x = 2$: $P(x) = x^3 - 7x^2 + 16x - 12$
Factorise the following cubic polynomial knowing that one of its roots is $x = 2$:
$P(x) = x^3 - 7x^2 + 16x - 12$
Find the integer roots of the polynomial $P(x) = x^3 + 2x^2 - 5x - 6$ using Ruffini's rule and express it in factorised form.
Find the integer roots of the polynomial $P(x) = x^3 + 2x^2 - 5x - 6$ using Ruffini's rule and express it in factorised form.
Complete factoring and multiple roots
Factoring "completely" means repeating common factor, special products, and synthetic division until every factor is of degree $1$ (or of degree $2$ with no real roots). If a root is repeated, it is a multiple root and its multiplicity is the number of times it appears.
State the roots of the polynomial $P(x) = 5(x+3)^2(x-1)(x^2+4)$ and their respective multiplicities. Justify why this polynomial has no other real roots.
State the roots of the polynomial $P(x) = 5(x+3)^2(x-1)(x^2+4)$ and their respective multiplicities. Justify why this polynomial has no other real roots.
Determine a fourth-degree polynomial, $P(x)$, that is monic (leading coefficient 1), has $x = -2$ as a double root, $x = 3$ as a simple root, and has no constant term.…
Determine a fourth-degree polynomial, $P(x)$, that is monic (leading coefficient 1), has $x = -2$ as a double root, $x = 3$ as a simple root, and has no constant term. Write it in its factorised form and in its general expanded form.
Fully factorise the polynomial $P(x) = x^6 - 2x^5 - 3x^4 + 8x^3 - 4x^2$. Clearly identify its real roots and the multiplicity of each of them.
Fully factorise the polynomial $P(x) = x^6 - 2x^5 - 3x^4 + 8x^3 - 4x^2$. Clearly identify its real roots and the multiplicity of each of them.
Biquadratic polynomials and repeated differences of squares
A biquadratic has only terms of degree $4$, $2$, and $0$: $ax^4+bx^2+c$. It is factored like a second-degree polynomial using the auxiliary variable $t=x^2$. Also, a difference of squares can be applied twice if the result is again one.
Write the polynomial $P(x) = x^4 - 81$ in factorised form by applying the difference of two squares identity successively.
Write the polynomial $P(x) = x^4 - 81$ in factorised form by applying the difference of two squares identity successively.
Fully factorise the biquadratic polynomial $P(x) = x^4 - 5x^2 + 4$.
Fully factorise the biquadratic polynomial $P(x) = x^4 - 5x^2 + 4$.
Rational roots: when the leading coefficient is not 1
If the leading coefficient is not $1$, then besides integers you also have to try fractions $\frac{p}{q}$, with $p$ a divisor of the constant term and $q$ a divisor of the leading coefficient.
Factorise the polynomial $P(x) = 3x^4 + 5x^3 - 17x^2 - 13x + 6$. Note that some of its roots may be fractional numbers.
Factorise the polynomial $P(x) = 3x^4 + 5x^3 - 17x^2 - 13x + 6$. Note that some of its roots may be fractional numbers.
A biologist models the growth and decline of a bacterial population using the polynomial $N(t) = -t^4 + 10t^3 - 35t^2 + 50t - 24$, where $t$ is the time in hours.…
A biologist models the growth and decline of a bacterial population using the polynomial $N(t) = -t^4 + 10t^3 - 35t^2 + 50t - 24$, where $t$ is the time in hours. Factorise the polynomial to determine the exact times at which the population becomes zero.
In an economics study, the marginal profit curve of a technology company fits the polynomial $B(x) = 6x^4 - 11x^3 - 26x^2 + 21x + 10$, where $x$ represents the R&D…
In an economics study, the marginal profit curve of a technology company fits the polynomial $B(x) = 6x^4 - 11x^3 - 26x^2 + 21x + 10$, where $x$ represents the R&D investment in millions of pesos. Factorize the polynomial to find the exact equilibrium points (including fractional roots).
Irreducible second-degree factors
Not every polynomial breaks into degree-$1$ factors. A degree-$2$ factor with a negative discriminant is irreducible over the reals, and that is where factoring stops. Sometimes, adding and subtracting an auxiliary term reveals two quadratic factors.
Prove that the polynomial $P(x) = x^4 + 4$ has no real roots. Then, factorise it as the product of two quadratic polynomials. Hint: Add and subtract $4x^2$ to the…
Prove that the polynomial $P(x) = x^4 + 4$ has no real roots. Then, factorise it as the product of two quadratic polynomials. Hint: Add and subtract $4x^2$ to the expression to force the appearance of a difference of two squares.
The design profile of a precision mechanical part is defined by the polynomial function $P(x) = x^5 - x^4 - 7x^3 + x^2 + 6x$. Factorise the polynomial to find the exact…
The design profile of a precision mechanical part is defined by the polynomial function $P(x) = x^5 - x^4 - 7x^3 + x^2 + 6x$. Factorise the polynomial to find the exact coordinates of the anchoring points of the part on the horizontal axis.
Algebraic fractions: simplifying and subtracting
Factoring is useful for working with algebraic fractions: you factor the numerator and denominator, cancel the common factors, and, if you have to add or subtract, you simplify each fraction before looking for the common denominator. You must always note the values of $x$ that make some denominator zero.
Simplify the following algebraic fraction by fully factorising the numerator and the denominator first: $\frac{x^3 - 3x^2 - 4x + 12}{x^3 - 5x^2 + 6x}$
Simplify the following algebraic fraction by fully factorising the numerator and the denominator first:
$\frac{x^3 - 3x^2 - 4x + 12}{x^3 - 5x^2 + 6x}$
Simplify the following algebraic expression, which represents the equivalent impedance in a complex circuit, by fully factorising each polynomial in the numerators and…
Simplify the following algebraic expression, which represents the equivalent impedance in a complex circuit, by fully factorising each polynomial in the numerators and denominators:
$\frac{2x^3 - 7x^2 + 7x - 2}{2x^3 - x^2 - 8x + 4} - \frac{x^2 - 1}{x^2 + x - 2}$
Factoring with parameters and applications
If a polynomial has a parameter and a divisibility condition, you first find the parameter (using the remainder theorem) and then factor with all the coefficients known. Factoring is also useful for finding where a quantity becomes zero.
Determine the value of the parameter $m$ so that the polynomial $P(x) = x^3 - mx^2 + 4x - 4$ is divisible by the binomial $x - 2$. Once the value of $m$ is found, fully…
Determine the value of the parameter $m$ so that the polynomial $P(x) = x^3 - mx^2 + 4x - 4$ is divisible by the binomial $x - 2$. Once the value of $m$ is found, fully factorise the polynomial over the set of real numbers.
The volume of a rectangular box as a function of its length $x$ is given by the polynomial expression $V(x) = x^3 - 7x^2 + 10x$. Factorise $V(x)$ to find the simplest…
The volume of a rectangular box as a function of its length $x$ is given by the polynomial expression $V(x) = x^3 - 7x^2 + 10x$. Factorise $V(x)$ to find the simplest algebraic expressions representing the three dimensions of the box.
Calculate the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 12x + 8$ is divisible by the polynomial $Q(x) = x^2 - 4$. Then,…
Calculate the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 12x + 8$ is divisible by the polynomial $Q(x) = x^2 - 4$. Then, fully factorise $P(x)$.