Factoring Polynomials
Wilkin wants to know when a polynomial function equals zero, but the polynomial has degree 4. If he writes it as $P(x)=2x(x+5)(x-4)^2$, he sees right away that $P(x)=0$ only when $x=0$, $x=-5$, or $x=4$. Factoring means taking the polynomial apart into its pieces. On this page we put all the techniques (common factor, special products, synthetic division, and rational zeros) into one procedure, and we see how many roots a polynomial can have and what kind they are.
Common factor and special products
Factoring always starts the same way: pull out all the common factor you can. After that, if what remains is a special product (for example, a difference of squares) or a quadratic trinomial, you solve it.
Another tool is the remainder theorem: the remainder when dividing $P(x)$ by $(x-a)$ is $P(a)$. Therefore, if $P(a)=0$, then $(x-a)$ is a factor of $P(x)$ (the factor theorem).
Why? When dividing, $P(x)=(x-a)Q(x)+R$ with $R$ constant. Substituting $x=a$ gives $P(a)=R$. If $R=0$, the factor $(x-a)$ divides evenly.
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.
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.
Fundamental theorem of algebra and multiplicity
A root that repeats is a multiple root; its multiplicity is the number of times it appears as a factor. If $r_1,\dots,r_n$ are all the roots of $P(x)$, of degree $n$ and leading coefficient $c$:
$$P(x)=c(x-r_1)(x-r_2)\cdots(x-r_n)$$A degree-2 factor such as $x^2+4$ has a negative discriminant: its two roots are complex conjugates ($\pm 2i$), not real.
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.
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.
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.
Biquadratics and successive differences of squares
A biquadratic polynomial has only terms of degree $4$, $2$, and $0$: $ax^4+bx^2+c$. It is factored like a quadratic, letting $t=x^2$. And a difference of squares can be applied twice if the result is again a difference of squares.
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 $P(x) = x^4 - 5x^2 + 4$.
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.
Rational zeros: leading coefficient other than 1
If the leading coefficient is not $1$, you must also test fractions: the candidates are $\frac{p}{q}$ 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.
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).
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.
Irreducible factors of degree 2
Not every polynomial breaks down into degree-$1$ factors with real coefficients. A degree-$2$ factor with a negative discriminant is irreducible over $\mathbb{R}$: its roots are two complex conjugates, and the real factorization stops there. (Over the complex numbers it would break down: $x^2+4=(x-2i)(x+2i)$.)
Sometimes, adding and subtracting a term lets you see two degree-2 factors where none seemed to exist.
The trajectory of a racing drone in an exhibition section follows the polynomial profile $h(x) = 2x^4 - 14x^3 + 30x^2 - 18x$, where $x$ is the horizontal distance in…
The trajectory of a racing drone in an exhibition section follows the polynomial profile $h(x) = 2x^4 - 14x^3 + 30x^2 - 18x$, where $x$ is the horizontal distance in meters and $h(x)$ is the height in meters. Factorize $h(x)$ to find the exact points where the drone touches the ground ($h(x) = 0$).
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.
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.
Factoring with a parameter
When the polynomial has a parameter and a divisibility condition, you first find the parameter with the remainder theorem and then factor with all the coefficients now known. It can also be used the other way around: from known values of $P$ you rebuild the polynomial.
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.
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 $P(x)$.
A cubic polynomial $P(x)$ satisfies $P(1) = 0$, $P(-2) = 0$, and when divided by the binomial $(x-2)$ the remainder of the division is $16$. Furthermore, its constant…
A cubic polynomial $P(x)$ satisfies $P(1) = 0$, $P(-2) = 0$, and when divided by the binomial $(x-2)$ the remainder of the division is $16$. Furthermore, its constant term is known to be $4$. Find the factorised expression for $P(x)$.
Algebraic fractions and applications
Factoring is used to simplify algebraic fractions, to operate with them (the common denominator comes from the factored denominators), and to interpret models.
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.
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.
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:
$\frac{2x^3 - 7x^2 + 7x - 2}{2x^3 - x^2 - 8x + 4} - \frac{x^2 - 1}{x^2 + x - 2}$
The outline of the vault of an avant-garde building is given by the geometric intersection of the curves $f(x) = x^4 - 2x^3$ and $g(x) = 7x^2 - 8x - 12$. Factorise the…
The outline of the vault of an avant-garde building is given by the geometric intersection of the curves $f(x) = x^4 - 2x^3$ and $g(x) = 7x^2 - 8x - 12$. Factorise the difference polynomial $D(x) = f(x) - g(x)$ to determine the $x$ coordinates of the pillars where both structures join.
Back to the beginning: with $P(x)=2x(x+5)(x-4)^2$ the roots can be read directly: $0$, $-5$, and $4$ (double).