Factoring Polynomials
Emma's design team is building a shipping box. The volume is $V(x) = x^3 + 3x^2 - 10x$ cubic inches, and she needs the length, width, and height. The answer is hiding inside the polynomial, and factoring is how you find it. A factored polynomial like $P(x) = 2x(x+5)(x-4)^2$ tells you at a glance that $P(x) = 0$ exactly when $x = 0$, $x = -5$, or $x = 4$. In this lesson you combine every factoring tool you know into one systematic procedure, and add the new cases: repeated zeros, fractional zeros, and quadratic factors with no real zeros.
Common factor, special products, and trinomials
Factoring always starts the same way: pull out the greatest common factor (GCF). Then look at what is left and ask whether it is a special product, a trinomial you can factor, or something you can group.
Trinomials and grouping
For a trinomial $ax^2 + bx + c$ that is hard to factor by inspection, solve $ax^2 + bx + c = 0$ with the quadratic formula. If the zeros are $r_1$ and $r_2$, then $ax^2 + bx + c = a(x - r_1)(x - r_2)$.
For four terms, try factoring by grouping: factor each pair of terms, then pull out the common binomial.
Factor out the common factor and use algebraic identities to factorize the following polynomial as much as possible: $P(x) = 3x^5 - 12x^3$
Factor out the common factor and use algebraic identities to factorize the following polynomial as much as possible:
$P(x) = 3x^5 - 12x^3$
Factorize the polynomial $P(x) = 2x^2 + 5x - 3$ by previously solving the associated quadratic equation.
Factorize the polynomial $P(x) = 2x^2 + 5x - 3$ by previously solving the associated quadratic equation.
Factorize the polynomial $P(x) = (x^2 - 4)^2 - (x + 2)^2$ using standard identities and extracting common factors, without needing to expand the whole polynomial first.
Factorize the polynomial $P(x) = (x^2 - 4)^2 - (x + 2)^2$ using standard identities and extracting common factors, without needing to expand the whole polynomial first.
Zeros and multiplicity
Factoring "completely" means repeating the steps until every factor is linear or a quadratic with no real zeros. If a factor $(x - r)$ appears more than once, $r$ is a repeated zero, and the number of times it appears is its multiplicity.
The multiplicity also shows up on the graph. At a zero of odd multiplicity the graph crosses the $x$-axis. At a zero of even multiplicity it touches the axis and turns around. Here, $-1$ has multiplicity 2 and $2$ has multiplicity 1.
Indicate what the roots of the polynomial $P(x) = 5(x+3)^2(x-1)(x^2+4)$ are and their respective multiplicities. Justify why this polynomial has no more real roots.
Indicate what the roots of the polynomial $P(x) = 5(x+3)^2(x-1)(x^2+4)$ are and their respective multiplicities. Justify why this polynomial has no more real roots.
Determine a fourth-degree polynomial, $P(x)$, which is monic (leading coefficient 1), has $x = -2$ as a double root, $x = 3$ as a simple root, and lacks a constant term.…
Determine a fourth-degree polynomial, $P(x)$, which is monic (leading coefficient 1), has $x = -2$ as a double root, $x = 3$ as a simple root, and lacks a constant term. Write it in its factorized form and in its expanded general form.
The Factor Theorem and synthetic division
How do you find a zero of a cubic or quartic? You test values. The Remainder Theorem makes testing quick, and the Factor Theorem turns a zero into a factor.
Once you know a zero $a$, synthetic division finds the quotient with very little work. Write the coefficients, bring down the first one, then repeat: multiply by $a$, add to the next coefficient. The last number is the remainder. It is the same arithmetic as long division by $x - a$, just without writing the variables. Some problems call this Ruffini's rule.
Check, using the Remainder Theorem, if $x = -1$ is a root of the polynomial $P(x) = x^4 + 3x^3 - 2x^2 + x + 7$. If so, extract the corresponding factor.
Check, using the Remainder Theorem, if $x = -1$ is a root of the polynomial $P(x) = x^4 + 3x^3 - 2x^2 + x + 7$. If so, extract the corresponding factor.
Factorize the following third-degree polynomial knowing that one of its roots is $x = 2$: $P(x) = x^3 - 7x^2 + 16x - 12$
Factorize the following third-degree 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 its factorized form.
Find the integer roots of the polynomial $P(x) = x^3 + 2x^2 - 5x - 6$ using Ruffini's rule and express it in its factorized form.
Quadratic-form polynomials and repeated differences of squares
A polynomial $ax^4 + bx^2 + c$ is quadratic in form: it only has terms of degree $4$, $2$, and $0$, so $x^4 = (x^2)^2$. Some problems call these biquadratic polynomials. Replace $x^2$ with a new variable $t$, factor the quadratic in $t$, then put $x^2$ back. If a difference of squares appears again after the first step, apply it a second time.
Write the polynomial $P(x) = x^4 - 81$ in factorized form by applying the difference of two squares identity successively.
Write the polynomial $P(x) = x^4 - 81$ in factorized form by applying the difference of two squares identity successively.
Completely factorize the biquadratic polynomial $P(x) = x^4 - 5x^2 + 4$.
Completely factorize the biquadratic polynomial $P(x) = x^4 - 5x^2 + 4$.
The Rational Root Theorem
If the leading coefficient is not $1$, the zeros can be fractions. The Rational Root Theorem gives you a finite list of candidates to test.
Factorize the polynomial $P(x) = 3x^4 + 5x^3 - 17x^2 - 13x + 6$. Bear in mind that some of its roots may be fractional numbers.
Factorize the polynomial $P(x) = 3x^4 + 5x^3 - 17x^2 - 13x + 6$. Bear in mind that some of its roots may be fractional numbers.
A biologist models the growth and decline of a bacteria 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 bacteria population using the polynomial $N(t) = -t^4 + 10t^3 - 35t^2 + 50t - 24$, where $t$ is the time in hours. Factorize the polynomial to determine at what exact times 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…
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 investment in R&D in millions of dollars. Factorize the polynomial to find the exact break-even points (including fractional roots).
A complete strategy for higher-degree polynomials
For polynomials of degree 4, 5, or 6, combine every tool. Follow the same order each time, and repeat steps 2 and 3 on each quotient.
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 feet and $h(x)$ is the height in feet. Factorize $h(x)$ to find the exact points where the drone touches the ground ($h(x) = 0$).
The profile design of a precision mechanical part is defined by the polynomial function $P(x) = x^5 - x^4 - 7x^3 + x^2 + 6x$. Factorize the polynomial to find the exact…
The profile design of a precision mechanical part is defined by the polynomial function $P(x) = x^5 - x^4 - 7x^3 + x^2 + 6x$. Factorize the polynomial to find the exact coordinates of the part's anchor points on the horizontal axis.
Completely factorize the polynomial $P(x) = x^6 - 2x^5 - 3x^4 + 8x^3 - 4x^2$. Clearly identify its real roots and the multiplicity of each one.
Completely factorize the polynomial $P(x) = x^6 - 2x^5 - 3x^4 + 8x^3 - 4x^2$. Clearly identify its real roots and the multiplicity of each one.
Irreducible quadratic factors
Not every polynomial breaks into linear factors. A quadratic factor with a negative discriminant has no real zeros. It is irreducible over the real numbers, and the factorization stops there. Over the complex numbers it keeps going: $x^2 + 9 = (x + 3i)(x - 3i)$. Complex zeros of a real polynomial come in conjugate pairs, and that is why real polynomials split into linear and irreducible quadratic factors.
Sometimes you can find quadratic factors that are not obvious. Adding and subtracting a clever term completes a square and creates a difference of squares.
Show that the polynomial $P(x) = x^4 + 4$ has no real roots. Then, factorize it as the product of two second-degree polynomials. Hint: Add and subtract $4x^2$ to the…
Show that the polynomial $P(x) = x^4 + 4$ has no real roots. Then, factorize it as the product of two second-degree polynomials. Hint: Add and subtract $4x^2$ to the expression to force the appearance of a difference of two squares.
Factoring with unknown parameters
When a polynomial has an unknown coefficient and a divisibility condition, use the Factor Theorem first to find the unknown. Then factor with all the coefficients known.
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,…
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, completely factorize the polynomial in 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, factorize $P(x)$ completely.
A third-degree 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, it is…
A third-degree 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, it is known that its constant term is $4$. Find the factorized expression of $P(x)$.
Applications: dimensions, break-even points, and intersections
Factoring turns a model into answers. A volume formula factors into dimensions. A profit or height model equal to zero gives break-even points or landing points. And where two curves $f$ and $g$ meet, $f(x) = g(x)$, so the zeros of $D(x) = f(x) - g(x)$ are the $x$-coordinates of the intersections.
The volume of a rectangular box as a function of its length $x$ is given by the polynomial expression $V(x) = x^3 - 7x^2 + 10x$. Factorize $V(x)$ to find the simplest…
The volume of a rectangular box as a function of its length $x$ is given by the polynomial expression $V(x) = x^3 - 7x^2 + 10x$. Factorize $V(x)$ to find the simplest algebraic expressions representing the three dimensions of the box.
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 $g(x) = 7x^2 - 8x - 12$. Factorize 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 $g(x) = 7x^2 - 8x - 12$. Factorize the difference polynomial $D(x) = f(x) - g(x)$ to determine the $x$ coordinates of the pillars where both structures meet.
Simplifying rational expressions
To simplify a fraction of polynomials, factor the numerator and the denominator completely, then cancel the common factors. The same idea is the first step when you add or subtract rational expressions: factor each denominator before you find the common denominator.
Simplify the following algebraic fraction by first factoring the numerator and the denominator completely: $\frac{x^3 - 3x^2 - 4x + 12}{x^3 - 5x^2 + 6x}$
Simplify the following algebraic fraction by first factoring the numerator and the denominator completely:
$\frac{x^3 - 3x^2 - 4x + 12}{x^3 - 5x^2 + 6x}$
Simplify the following algebraic expression, which represents the equivalent impedance in a complex circuit, by completely factoring each polynomial in the numerators…
Simplify the following algebraic expression, which represents the equivalent impedance in a complex circuit, by completely factoring each polynomial in the numerators and denominators:
$\frac{2x^3 - 7x^2 + 7x - 2}{2x^3 - x^2 - 8x + 4} - \frac{x^2 - 1}{x^2 + x - 2}$
Summary
Back to the start: factoring $x^3 + 3x^2 - 10x$ gave $x(x+5)(x-2)$, so Emma's box measures $x$ by $x+5$ by $x-2$ inches.