Factoring Polynomials and Synthetic Division
Jayden launches a toy rocket in his backyard. A model for its height in feet, $t$ seconds after launch, is $h(t) = -t^3 + 4t^2 + 12t$. When does the rocket land? You would have to solve $-t^3 + 4t^2 + 12t = 0$, and there is no quick formula for a cubic. But if the polynomial is written as a product of factors, the answer jumps out, because a product is zero only when one of its factors is zero. In this page you learn how to break a polynomial into factors, and the tool that makes it possible for higher degrees: synthetic division.
Factoring out the GCF
Every factoring problem starts the same way: pull out the greatest common factor (GCF). To find it, take the GCF of the coefficients, and for each variable the smallest exponent that appears in every term.
Factor out the common factor in the following polynomial, simplifying the expression fully: $P(x) = 15x^5 - 10x^4 + 20x^3$
Factor out the common factor in the following polynomial, simplifying the expression fully:
$P(x) = 15x^5 - 10x^4 + 20x^3$
Special products in reverse
After the GCF, look at what is left. If it matches one of the special products you learned for multiplying, you can factor it at once, just by running the pattern backward.
| Pattern | Factored form |
|---|---|
| $a^2 + 2ab + b^2$ (perfect square trinomial) | $(a + b)^2$ |
| $a^2 - 2ab + b^2$ (perfect square trinomial) | $(a - b)^2$ |
| $a^2 - b^2$ (difference of squares) | $(a + b)(a - b)$ |
For a perfect square trinomial, the first and last terms must be perfect squares, and the middle term must be $2ab$, twice the product of their square roots. For a difference of squares, there must be exactly two terms, both perfect squares, with a minus sign between them.
Special products are also a powerful tool for proofs. To show that an expression is always a multiple of $k$, factor it and look for $k$.
Factor the following polynomials by recognizing them as the expansion of algebraic identities (perfect square of a sum, perfect square of a difference, or difference of…
Factor the following polynomials by recognizing them as the expansion of algebraic identities (perfect square of a sum, perfect square of a difference, or difference of two squares):
- a) $A(x) = x^2 - 14x + 49$
- b) $B(x) = 9x^2 - 16$
A solar panel has a rectangular shape and its area is given by the polynomial $A(x) = x^4 - 81$. Factor the polynomial completely into irreducible polynomials with real…
A solar panel has a rectangular shape and its area is given by the polynomial $A(x) = x^4 - 81$. Factor the polynomial completely into irreducible polynomials with real coefficients to find the possible algebraic expressions for its base and height.
Using polynomial factorization and algebraic identities, prove algebraically that the difference between the squares of two consecutive odd numbers,…
Using polynomial factorization 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 $n$.
Synthetic division
For a polynomial of degree 3 or higher, the GCF and special products are rarely enough. The central tool is synthetic division, a fast way to divide $P(x)$ by a binomial $(x - a)$ using only the coefficients. Some problems call it Ruffini's rule; it is the same method.
Here is the idea, with $P(x) = x^3 + 2x^2 - 5x - 6$ divided by $(x - 2)$:
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Write the coefficients of $P(x)$ in a row, putting a $0$ for any missing power. Put $a$ (here $2$) to the left.
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Bring down the first coefficient.
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Multiply it by $a$, write the result under the next coefficient, and add.
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Repeat step 3 until you run out of coefficients.
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The last number is the remainder. The other bottom numbers are the coefficients of the quotient, which has degree one less than $P(x)$.
What if the divisor is not monic, like $2x - 1$? Rewrite it as $2\left(x - \tfrac{1}{2}\right)$, divide by $x - \tfrac12$, and then divide the quotient by $2$. The remainder does not change.
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by the binomial $(x - 2)$. State clearly which is the quotient polynomial and which is the…
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by the binomial $(x - 2)$. State clearly which is the quotient polynomial and which is the remainder of the division.
Perform the division of 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…
Perform the division of 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 subsequently adjusting the quotient obtained.
The Remainder Theorem and the Factor Theorem
Before building a whole synthetic division table for a possible root, you can test it much faster. Division gives $P(x) = (x - a)\,Q(x) + R$. Now substitute $x = a$: the term $(a - a)\,Q(a)$ becomes $0$, leaving $P(a) = R$.
The roots of a polynomial are the $x$-intercepts of its graph. The graph below shows $P(x) = x^3 + 3x^2 - 4x - 12$ crossing the $x$-axis at each root.
Without performing the division, use the remainder theorem to find the remainder when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.
Without performing the division, use the remainder theorem to find the remainder when dividing the polynomial $P(x) = 2x^4 - x^3 + 5x - 3$ by $(x + 1)$.
Use the factor theorem to check if $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your work.
Use the factor theorem to check if $x = 3$ is a root of the polynomial $P(x) = x^3 - 4x^2 + x + 6$. Justify your answer by showing your work.
Finding candidate roots
Guessing numbers at random is slow. Luckily, if the coefficients are integers, the possible rational roots can be listed.
The list gives candidates, not guarantees. You test each one with the Remainder Theorem, and when one works you divide by synthetic division.
It is known that the polynomial $P(x) = 2x^3 + x^2 - 13x + 6$ has at least one integer root. Check using the remainder theorem or Ruffini's rule 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. Check using the remainder theorem or Ruffini's rule that $x = 2$ is one of its roots. Next, factor the polynomial completely to find its other two roots (one of them is a fraction).
A team of engineers analyzes the stress distribution in a metal beam, which is modeled by the polynomial $P(x) = x^4 - x^3 - 7x^2 + x + 6$. Find all the roots of the…
A team of engineers analyzes the stress distribution in a metal beam, which is modeled 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.
Factoring completely
To factor completely means to keep going until every factor has degree 1, or degree 2 with no real roots. Use this order every time:
For a leftover quadratic $x^2 + bx + c$, look for two numbers that multiply to $c$ and add to $b$. For $ax^2 + bx + c$ with $a \neq 1$, use grouping (as in the example above) or the quadratic formula.
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 factor 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 factor it and find the other two roots.
Factor the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$ completely. To do this, first factor out the common factor and then find the roots of the resulting polynomial.
Factor the polynomial $P(x) = 2x^4 - 2x^3 - 12x^2$ completely. To do this, first factor out the common factor and then find the roots of the resulting polynomial.
A packaging company designs a cardboard box whose volume is given by the polynomial $V(x) = x^3 - 12x^2 + 36x$. Factor the polynomial fully to express the volume as the…
A packaging company designs a cardboard box whose volume is given by the polynomial $V(x) = x^3 - 12x^2 + 36x$. Factor the polynomial fully to express the volume as the product of three linear dimensions (length, width, and height).
Multiple roots
Sometimes the same factor appears more than once. If $(x - r)$ appears $k$ times, then $r$ is a root of multiplicity $k$. A root of multiplicity 1 is simple, multiplicity 2 is double, multiplicity 3 is triple.
Multiplicity also shows on the graph. At a root of odd multiplicity the graph crosses the $x$-axis; at a root of even multiplicity it just touches the axis and turns around. Here $x = -1$ is a double root and $x = 2$ is simple:
When you count roots with multiplicity, a polynomial of degree $n$ has at most $n$ real roots (exactly $n$ if all factors are linear).
Given the already factored polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly what all its roots are and specify the multiplicity of each one (whether they are…
Given the already factored polynomial $P(x) = 2x(x + 5)(x - 4)^2$, state directly what all its roots are and specify the multiplicity of each one (whether they are simple, double, etc.).
Factor the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$ completely. Bear in mind that it may contain multiple roots. State clearly what its roots are and the…
Factor the polynomial $P(x) = x^4 - 2x^3 - 3x^2 + 4x + 4$ completely. Bear in mind that it may contain multiple roots. State clearly what its roots are and the multiplicity of each one.
In a cryptography system, a security key is validated if it matches the roots of the polynomial $C(x) = x^5 - 4x^4 + 4x^3 + 2x^2 - 5x + 2$. Factor the polynomial…
In a cryptography system, a security key is validated if it matches the roots of the polynomial $C(x) = x^5 - 4x^4 + 4x^3 + 2x^2 - 5x + 2$. Factor the polynomial completely to discover all the roots and their respective multiplicities, thus managing to decipher the code.
Building a polynomial from its roots
Factoring works in reverse too. If you know the roots, the Factor Theorem hands you the factors, and then you multiply them out.
If a problem says the leading coefficient is $1$, then $a = 1$ and there is nothing to solve for. A double root means its factor appears twice: roots $1$ (double) and $-2$ give $(x - 1)^2(x + 2)$.
Construct a polynomial of degree 3 that has $x_1 = -2$, $x_2 = 0$, and $x_3 = 4$ as roots, and whose leading coefficient is $1$. Express the final result in expanded…
Construct a polynomial of degree 3 that has $x_1 = -2$, $x_2 = 0$, and $x_3 = 4$ as roots, and whose leading coefficient is $1$. Express the final result in expanded form (multiplying out the factors).
Construct a polynomial $P(x)$ of degree 3 that has $x_1 = 1$, $x_2 = -2$, and $x_3 = 3$ as roots, knowing also that its numerical value at $x = 0$ is $12$ (that is,…
Construct a polynomial $P(x)$ of degree 3 that has $x_1 = 1$, $x_2 = -2$, and $x_3 = 3$ as roots, knowing also that its numerical value at $x = 0$ is $12$ (that is, $P(0) = 12$). Express the final polynomial in expanded form.
Finding unknown coefficients
The theorems also solve for unknown coefficients. Each condition ("divisible by $x - a$", "remainder $R$") becomes an equation, found by substituting $x = a$.
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"Divisible by $(x - a)$" means $P(a) = 0$.
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"Remainder $R$ when divided by $(x - a)$" means $P(a) = R$.
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 $(x - 2)$.
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, factor the polynomial…
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, factor the polynomial $P(x)$ completely.
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 $(x + 2)$.
Factoring in the real world
Whenever a polynomial models a situation and you want to know when it equals zero, factor it. Two things to remember: factor out the GCF first, and then keep only the answers that make sense (time cannot be negative, a length must be positive).
Back to the start: the rocket lands after exactly $6$ seconds, and we found it without any cubic formula, just by factoring.
A water tank has the shape of a rectangular prism. Its volume in terms of a measurement $x$ is given by the polynomial $V(x) = x^3 + 5x^2 + 6x$. If the height of the…
A water tank has the shape of a rectangular prism. Its volume in terms of a measurement $x$ is given by the polynomial $V(x) = x^3 + 5x^2 + 6x$. If the height of the tank is exactly $x$, factor the polynomial to find the algebraic expressions that correspond to the dimensions of its rectangular base.
The height of a test drone, in feet, with respect to the time $t$ in seconds since it begins its descent, is modeled by the polynomial $H(t) = -t^3 + 8t^2 - 15t$. Factor…
The height of a test drone, in feet, with respect to the time $t$ in seconds since it begins its descent, is modeled by the polynomial $H(t) = -t^3 + 8t^2 - 15t$. Factor the polynomial to determine at what exact times the drone is at ground level ($H(t) = 0$).
The monthly profits of a company, in thousands of dollars, are given by the expression $B(x) = x^3 - 14x^2 + 49x$, where $x$ is the number of months elapsed since its…
The monthly profits of a company, in thousands of dollars, are given by the expression $B(x) = x^3 - 14x^2 + 49x$, where $x$ is the number of months elapsed since its creation. Factor the expression to find out in which month or months the company's profit is exactly zero.