Factoring and synthetic division
For someone solving equations, a factored polynomial is what an assembly diagram is for someone building furniture: it tells you where it "comes apart." If $P(x)=2x(x+5)(x-4)^2$, at a glance you know that $P(x)=0$ only when $x=0$, $x=-5$, or $x=4$. Here you will learn to factor using the common factor, special products, trinomials, and grouping; and, as an extension, you will see how synthetic division turns larger polynomials into a diagram like that.
Common factor and special products
Factoring always starts the same way: take out the greatest possible common factor. It is the GCF of the coefficients and, for each letter, the one that appears in all the terms with its smallest exponent. After that, check whether what is left is a special product.
| Special product | Factoring |
|---|---|
| Square of a binomial | $a^2+2ab+b^2=(a+b)^2$ |
| Square of a binomial | $a^2-2ab+b^2=(a-b)^2$ |
| Conjugate binomials | $a^2-b^2=(a+b)(a-b)$ |
| Trinomial $x^2+bx+c$ | $(x+p)(x+q)$ with $p+q=b$ and $p\cdot q=c$ |
A trinomial is a perfect square if two terms are squares and the other is twice the product of their roots. For example, $x^2-14x+49$: $x^2$ and $49=7^2$ are squares, and $2\cdot x\cdot 7=14x$. So $x^2-14x+49=(x-7)^2$.
Factorise out the common factor in the following polynomial, simplifying the expression as much as possible: $P(x) = 15x^5 - 10x^4 + 20x^3$
Factorise out the common factor in the following polynomial, simplifying the expression as much as possible:
$P(x) = 15x^5 - 10x^4 + 20x^3$
Factorise the following polynomials by recognising algebraic identities (square of a sum, square of a difference, or difference of two squares): a)…
Factorise the following polynomials by recognising algebraic identities (square of a sum, square of a difference, or difference of two squares):
- a) $A(x) = x^2 - 14x + 49$
- b) $B(x) = 9x^2 - 16$
A water tank is in the shape of a rectangular prism. Its volume as a function of a measurement $x$ is given by the polynomial $V(x) = x^3 + 5x^2 + 6x$. If the height of…
A water tank is in the shape of a rectangular prism. Its volume as a function of a measurement $x$ is given by the polynomial $V(x) = x^3 + 5x^2 + 6x$. If the height of the tank is exactly $x$, factorise the polynomial to find the algebraic expressions corresponding to the dimensions of its rectangular base.
Factoring by grouping
If the polynomial has four terms, it is sometimes helpful to group them in pairs, take out the common factor in each group, and look for a repeated parenthesis.
And combining everything, a difference of squares can repeat: $x^4-81=(x^2+9)(x^2-9)=(x^2+9)(x+3)(x-3)$. The factor $x^2+9$ can no longer be factored over the real numbers.
A packaging company designs a cardboard box whose volume is given by the polynomial $V(x) = x^3 - 12x^2 + 36x$. Factorise the polynomial fully to express the volume as…
A packaging company designs a cardboard box whose volume is given by the polynomial $V(x) = x^3 - 12x^2 + 36x$. Factorise the polynomial fully to express the volume as the product of three linear dimensions (length, width and height).
A solar panel is rectangular in shape and its area is given by the polynomial $A(x) = x^4 - 81$. Fully factorise the polynomial into irreducible polynomials with real…
A solar panel is rectangular in shape and its area is given by the polynomial $A(x) = x^4 - 81$. Fully factorise the polynomial into irreducible polynomials with real coefficients to find the possible algebraic expressions for its base and height.
Using polynomial factorisation and algebraic identities, prove algebraically that the difference between the squares of two consecutive odd numbers,…
Using polynomial factorisation and algebraic identities, prove algebraically that the difference between the squares of two consecutive odd numbers, $(2n+1)^2 - (2n-1)^2$, is always a multiple of $8$ for any integer $n$.
From the factors to the zeros
Factoring is useful for knowing when a polynomial equals zero. If a product is $0$, one of its factors is $0$. That is why, in $2x(x+5)(x-4)^2=0$, it is enough to set each factor equal to zero.
Given the already factorised polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly all of its roots and specify the multiplicity of each root (whether they are simple,…
Given the already factorised polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly all of its roots and specify the multiplicity of each root (whether they are simple, double, etc.).
The height of a roller coaster car, in meters, relative to time $t$ in seconds from the start of a drop, is modeled by the polynomial $H(t) = -t^3 + 8t^2 - 15t$. Factor…
The height of a roller coaster car, in meters, relative to time $t$ in seconds from the start of a drop, is modeled by the polynomial $H(t) = -t^3 + 8t^2 - 15t$. Factor the polynomial to determine the exact times at which the car is at ground level ($H(t) = 0$).
The number of visits to a website (in thousands) during month $x$ since its launch is given by the expression $V(x) = x^3 - 14x^2 + 49x$. Factorise the expression to…
The number of visits to a website (in thousands) during month $x$ since its launch is given by the expression $V(x) = x^3 - 14x^2 + 49x$. Factorise the expression to find in which month or months the number of visits was exactly zero.
Extension: synthetic division (Ruffini's rule)
For polynomials of degree $3$ or higher, the common factor and special products almost never suffice. The tool is synthetic division (Ruffini's rule): if $a$ is a root, it divides $P(x)$ by $(x-a)$ working only with the coefficients and gives a quotient of one degree less.
You bring down the first coefficient; each result is multiplied by $a$ and added to the next coefficient. The last number is the remainder.
If the divisor is $(2x-3)$, divide by $(x-\tfrac32)$ and then divide the quotient by $2$, because $2x-3=2\left(x-\tfrac32\right)$. The remainder does not change.
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by the binomial $(x - 2)$. Clearly state the quotient polynomial and the remainder of the…
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by the binomial $(x - 2)$. Clearly state the quotient polynomial and the remainder of the division.
Given that $x = 1$ is one of the roots of the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$, use Ruffini's rule to factorise it and find the other two roots.
Given that $x = 1$ is one of the roots of the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$, use Ruffini's rule to factorise it and find the other two roots.
Divide the polynomial $P(x) = 4x^3 - 8x^2 + 5x - 1$ by the binomial $(2x - 3)$. Hint: You can use Ruffini's rule by dividing by $(x - \frac{3}{2})$ and then adjusting…
Divide the polynomial $P(x) = 4x^3 - 8x^2 + 5x - 1$ by the binomial $(2x - 3)$. Hint: You can use Ruffini's rule by dividing by $(x - \frac{3}{2})$ and then adjusting the quotient obtained.
Extension: the remainder theorem and the factor theorem
Before building the table with just any candidate, it helps to know whether it is worth it. The remainder theorem is useful for that.
It is also useful for finding parameters: if $x^3+kx^2-3x+8$ must leave remainder $10$ when divided by $(x-2)$, you set up $P(2)=10$ and solve for $k$.
Without performing the division, use the remainder theorem to calculate the remainder obtained when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.
Without performing the division, use the remainder theorem to calculate the remainder obtained when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.
Use the factor theorem to check whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your calculations.
Use the factor theorem to check whether $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your calculations.
Find the value of the parameter $k$ so that the polynomial $P(x) = x^3 + kx^2 - 3x + 8$ gives a remainder of $10$ when divided by $(x - 2)$.
Find the value of the parameter $k$ so that the polynomial $P(x) = x^3 + kx^2 - 3x + 8$ gives a remainder of $10$ when divided by $(x - 2)$.
Extension: integer root candidates
Trying numbers at random is inefficient. If the coefficients are integers, the integer roots are among the divisors of the constant term.
Calculate the value of $m$ so that the polynomial $P(x) = x^3 - 2x^2 + mx + 6$ is exactly divisible by $(x - 3)$. Once the value of $m$ is found, fully factorise the…
Calculate the value of $m$ so that the polynomial $P(x) = x^3 - 2x^2 + mx + 6$ is exactly divisible by $(x - 3)$. Once the value of $m$ is found, fully factorise the polynomial $P(x)$.
Determine the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 4x + 4$ is simultaneously divisible by $(x - 1)$ and by $(x + 2)$.
Determine the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 + ax^3 + bx^2 - 4x + 4$ is simultaneously divisible by $(x - 1)$ and by $(x + 2)$.
It is known that the polynomial $P(x) = 2x^3 + x^2 - 13x + 6$ has at least one integer root. Use the remainder theorem or Ruffini's rule to check that $x = 2$ is one of…
It is known that the polynomial $P(x) = 2x^3 + x^2 - 13x + 6$ has at least one integer root. Use the remainder theorem or Ruffini's rule to check that $x = 2$ is one of its roots. Next, fully factorise the polynomial to find its other two roots (one of which is a fraction).
A team of engineers analyses the stress distribution in a metal beam, which is modelled by the polynomial $P(x) = x^4 - x^3 - 7x^2 + x + 6$. Find all the roots of the…
A team of engineers analyses the stress distribution in a metal beam, which is modelled by the polynomial $P(x) = x^4 - x^3 - 7x^2 + x + 6$. Find all the roots of the polynomial to determine the exact points where the stress is zero.
Extension: complete factoring and multiple roots
To factor completely is to repeat the common factor, special products, and synthetic division until every factor has degree $1$ (or degree $2$ with no real roots). If a root repeats, it is a multiple root.
To build a polynomial from its roots, multiply the factors $(x-r)$ and, if you are given a value such as $P(0)$, adjust the leading coefficient $c$.
Construct a polynomial of degree 3 that has roots $x_1 = -2$, $x_2 = 0$ and $x_3 = 4$, and whose leading coefficient is $1$. Express the final result in expanded form…
Construct a polynomial of degree 3 that has roots $x_1 = -2$, $x_2 = 0$ and $x_3 = 4$, and whose leading coefficient is $1$. Express the final result in expanded form (by multiplying out the factors).
Fully factorise the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$. To do this, first factorise out the common factor and then find the roots of the resulting polynomial.
Fully factorise the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$. To do this, first factorise out the common factor and then find the roots of the resulting polynomial.
Construct a polynomial $P(x)$ of degree 3 that has roots $x_1 = 1$, $x_2 = -2$ and $x_3 = 3$, knowing also that its value at $x = 0$ is $12$ (that is, $P(0) = 12$).…
Construct a polynomial $P(x)$ of degree 3 that has roots $x_1 = 1$, $x_2 = -2$ and $x_3 = 3$, knowing also that its value at $x = 0$ is $12$ (that is, $P(0) = 12$). Express the final polynomial in expanded form.
Fully factorise the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$. Note that it may contain multiple roots. Clearly state its roots and the multiplicity of each one.
Fully factorise the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$. Note that it may contain multiple roots. Clearly state its roots and the multiplicity of each one.
In a cryptography system, a security key is validated if it matches the roots of the polynomial $C(x) = x^5 - 4x^4 + 4x^3 + 2x^2 - 5x + 2$. Fully factorise the…
In a cryptography system, a security key is validated if it matches the roots of the polynomial $C(x) = x^5 - 4x^4 + 4x^3 + 2x^2 - 5x + 2$. Fully factorise the polynomial to find all the roots and their respective multiplicities, thereby deciphering the code.
Go back to the beginning: with the factoring $2x(x+5)(x-4)^2$ you read the roots $0$, $-5$, and $4$ (the last one double) without solving anything.