Polynomials
You are making gift boxes for a school bake sale from a sheet of cardboard 18 inches by 12 inches. You cut a square of side $x$ out of each corner and fold up the sides. How much will the box hold? The volume turns out to be a polynomial in $x$, and the tools in this page (dividing, factoring, finding zeros) are exactly what you need to answer questions about it. We will come back to the box at the end.
Polynomials and their operations
A polynomial in $x$ is a sum of terms $a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$. The numbers $a_i$ are the coefficients and the exponents are whole numbers. The degree is the highest exponent with a nonzero coefficient, $a_n$ is the leading coefficient, and $a_0$ is the constant term. Writing the terms from highest to lowest degree is standard form.
To add or subtract polynomials, combine like terms (same power of $x$). To multiply, distribute: every term of the first polynomial times every term of the second, then combine like terms.
Let the polynomials be $P(x) = 2x^3 - 5x^2 + 3x - 1$ and $Q(x) = -x^3 + 4x^2 - 2x + 5$. Calculate the resulting polynomial from the operation $2P(x) - Q(x)$ and express…
Let the polynomials be $P(x) = 2x^3 - 5x^2 + 3x - 1$ and $Q(x) = -x^3 + 4x^2 - 2x + 5$. Calculate the resulting polynomial from the operation $2P(x) - Q(x)$ and express the result ordered from highest to lowest degree.
Polynomial long division
Dividing a polynomial $P(x)$ by a polynomial $d(x)$ works like long division with numbers. You get a quotient $q(x)$ and a remainder $r(x)$ (some problems write the quotient as $C(x)$), and
$$P(x) = d(x)\cdot q(x) + r(x),$$where the degree of $r(x)$ is less than the degree of $d(x)$ (or $r(x)=0$).
When the dividend contains a parameter such as $k$, just carry $k$ along like any other coefficient. The remainder will then contain $k$, and conditions on the remainder (for example, "it must be a constant") give you an equation for $k$.
Perform the following polynomial division and clearly indicate which is the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$:…
Perform the following polynomial division and clearly indicate which is the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$:
$(x^4 - 3x^3 + 2x^2 - x + 5) : (x^2 - x + 1)$
In the division of the polynomial $P(x) = 2x^4 - x^3 + kx^2 - 3x + 5$ by $Q(x) = x^2 - x + 2$, find the value of the parameter $k$ so that the remainder of the division…
In the division of the polynomial $P(x) = 2x^4 - x^3 + kx^2 - 3x + 5$ by $Q(x) = x^2 - x + 2$, find the value of the parameter $k$ so that the remainder of the division is a constant number (that is, a polynomial of degree 0). What is the exact value of that remainder?
Synthetic division
When the divisor has the form $x-a$, you do not need the full long division. Synthetic division uses only the coefficients and the number $a$, and it turns the subtractions into additions. Some problems call this shortcut Ruffini's rule; it is the same method.
Use Ruffini's rule to perform the division of the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by the binomial $x + 2$. Indicate the quotient and the remainder.
Use Ruffini's rule to perform the division of the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by the binomial $x + 2$. Indicate the quotient and the remainder.
The Remainder Theorem and the Factor Theorem
Before dividing, it helps to ask whether the division is even worth doing. Two theorems answer that.
The graph shows what a zero means: $P(a)=0$ exactly where the curve touches the $x$-axis.
Apply the Remainder Theorem to calculate the remainder of the division of the polynomial $P(x) = x^5 - 3x^4 + 2x^2 - 5x + 8$ by $x - 3$, without needing to perform the…
Apply the Remainder Theorem to calculate the remainder of the division of the polynomial $P(x) = x^5 - 3x^4 + 2x^2 - 5x + 8$ by $x - 3$, without needing to perform the full division.
Using the Factor Theorem, check if the values $x = 2$ and $x = -1$ are roots of the polynomial $P(x) = x^3 - 2x^2 - x + 2$. Justify your answer by showing the…
Using the Factor Theorem, check if the values $x = 2$ and $x = -1$ are roots of the polynomial $P(x) = x^3 - 2x^2 - x + 2$. Justify your answer by showing the calculations.
Factoring polynomials
To factor completely, follow this order:
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Factor out the greatest common factor (GCF).
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Look for a special product pattern.
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Try trinomial factoring, grouping, or a substitution (quadratic form).
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If nothing works, look for a zero and use the Factor Theorem (next section).
Factor out the common factor and use algebraic identities to completely factorize the polynomial $P(x) = 3x^3 - 12x$.
Factor out the common factor and use algebraic identities to completely factorize the polynomial $P(x) = 3x^3 - 12x$.
Completely factorize the fourth-degree polynomial $P(x) = x^4 - 5x^2 + 4$. From its factorized form, indicate all its real roots.
Completely factorize the fourth-degree polynomial $P(x) = x^4 - 5x^2 + 4$. From its factorized form, indicate all its real roots.
Finding zeros and factoring higher-degree polynomials
When no pattern appears, you can search for zeros. If a polynomial has integer coefficients, the Rational Root Theorem says that any rational zero $\tfrac{p}{q}$ has $p$ dividing the constant term and $q$ dividing the leading coefficient. When the leading coefficient is $1$, the rational zeros are integers that divide the constant term.
The plan:
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List the divisors of the constant term (with $\pm$).
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Test candidates with the Remainder Theorem or synthetic division until one gives remainder $0$.
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Use the quotient, which has lower degree, and repeat. Finish with factoring or the quadratic formula.
To solve a polynomial equation, first move everything to one side so that it reads $P(x)=0$, then factor.
If a quadratic factor does not factor nicely, use the quadratic formula. When the discriminant $b^2-4ac$ is negative, there are no real zeros (you met complex zeros $a\pm bi$ in Integrated Math II).
For example, $x^4-81=(x-3)(x^3+3x^2+9x+27)$. It also equals $(x^2-9)(x^2+9)$, which shows how one expression can be factored in more than one way.
A cylindrical container is inscribed inside a sphere of radius $R = 10 \text{ cm}$. The volume of the cylinder as a function of its semi-height $x$ is…
A cylindrical container is inscribed inside a sphere of radius $R = 10 \text{ cm}$. The volume of the cylinder as a function of its semi-height $x$ is $V(x) = 2\pi x (100 - x^2)$. It is desired that the volume be exactly $768\pi \text{ cm}^3$. Formulate the polynomial equation $P(x) = 0$, find its integer root using Ruffini's rule and factorize the resulting polynomial to find all possible semi-heights $x$ that satisfy the geometric condition.
Prove, using the Factor Theorem, that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n \ge 1$. Next, apply this property to…
Prove, using the Factor Theorem, that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n \ge 1$. Next, apply this property to extract the obvious root and completely factorize the polynomial $P(x) = x^5 - 32$ in the set of real numbers.
Simplifying and multiplying rational expressions
A rational expression is a quotient of two polynomials. It behaves like a fraction of numbers: you simplify by cancelling common factors. Every value of $x$ that makes a denominator $0$ is an excluded value, and it stays excluded even after simplifying.
To multiply, factor everything first, then cancel any numerator factor against any denominator factor. To divide, multiply by the reciprocal of the second expression.
Simplify the following algebraic fraction as much as possible, previously factorizing the numerator and the denominator: $\frac{x^2 - 9}{x^2 + 6x + 9}$
Simplify the following algebraic fraction as much as possible, previously factorizing the numerator and the denominator:
$\frac{x^2 - 9}{x^2 + 6x + 9}$
Perform the following multiplication of algebraic fractions and simplify the result: $\frac{2x}{x-1} \cdot \frac{x^2 - 1}{4x^2}$
Perform the following multiplication of algebraic fractions and simplify the result:
$\frac{2x}{x-1} \cdot \frac{x^2 - 1}{4x^2}$
The area of a specially designed window in a building is given by the expression $A(x) = \frac{x^3 - 8}{x^2 - 4}$. Simplify this algebraic fraction by previously…
The area of a specially designed window in a building is given by the expression $A(x) = \frac{x^3 - 8}{x^2 - 4}$. Simplify this algebraic fraction by previously factorizing the numerator (difference of two cubes) and the denominator. Then, evaluate the area when $x = 3$ feet.
Adding, subtracting, and complex fractions
To add or subtract rational expressions, find the least common denominator (LCD): factor each denominator, then take every factor the greatest number of times it appears. Rewrite each fraction over the LCD, combine the numerators, and simplify.
A complex fraction has fractions in its numerator or denominator. Treat the big fraction bar as division: multiply by the reciprocal, factor, and cancel.
When a problem asks you to find $x$ so that a simplified expression equals a number, set up the equation, clear the denominator, and solve. Check the discriminant if the equation turns out to be quadratic: it may have no real solution.
In an electrical circuit, the equivalent resistance $R_{eq}$ is given by the following expression of algebraic fractions:…
In an electrical circuit, the equivalent resistance $R_{eq}$ is given by the following expression of algebraic fractions:
$R_{eq}(x) = \frac{2}{x^2-1} - \frac{1}{x-1} + \frac{x}{x+1}$
Perform the indicated operations, calculate the least common multiple of the denominators and simplify the resulting fraction as much as possible. What is the value of the resistance if $x = 3 \ \Omega$?
In the design of a gear mechanism, the transmission ratio is modeled by the following complex algebraic expression:…
In the design of a gear mechanism, the transmission ratio is modeled by the following complex algebraic expression:
$T(x) = \frac{\frac{x^2 - 4}{x^3 + 8}}{\frac{x^2 - 2x}{x^2 - 2x + 4}}$
Simplify $T(x)$ as much as possible by previously factorizing all numerators and denominators. What mathematical (and therefore physical) problem occurs with the transmission ratio if $x = 2$?
Given the algebraic fraction that models the signal-to-noise ratio in a telecommunications network: $S(x) = \frac{x^4 - 2x^3 - 7x^2 + 8x + 12}{x^3 - 5x^2 + 2x + 8}$…
Given the algebraic fraction that models the signal-to-noise ratio in a telecommunications network:
$S(x) = \frac{x^4 - 2x^3 - 7x^2 + 8x + 12}{x^3 - 5x^2 + 2x + 8}$
Completely factorize the numerator and the denominator using Ruffini's rule and solving quadratic equations. Simplify the fraction as much as possible and determine, if it exists, for what real value of $x$ the simplified fraction is equal to $3$.
The Binomial Theorem
Expanding $(a+b)^n$ by hand gets long fast. The coefficients follow a pattern, Pascal's triangle: each number is the sum of the two numbers above it.
Completely expand the power of the binomial $(2x - y)^4$ using the Binomial Theorem or Pascal's triangle.
Completely expand the power of the binomial $(2x - y)^4$ using the Binomial Theorem or Pascal's triangle.
In a study of genetic combinations, the distribution of a certain probability is related to the expansion of the binomial $\left(x^2 - \frac{1}{x}\right)^6$. Find the…
In a study of genetic combinations, the distribution of a certain probability is related to the expansion of the binomial $\left(x^2 - \frac{1}{x}\right)^6$. Find the exact numerical value of the constant term (the term that does not depend on $x$) in this expansion.
In the expansion of the binomial $(x + 2)^n$, it is known that the coefficient of the term containing $x^{n-2}$ is exactly $180$. Set up the corresponding equation using…
In the expansion of the binomial $(x + 2)^n$, it is known that the coefficient of the term containing $x^{n-2}$ is exactly $180$. Set up the corresponding equation using the Binomial Theorem, find the value of the natural number $n$ and calculate the constant term of said expansion.
Polynomials as volume models
Now back to the bake-sale box. Geometry problems often produce polynomials: lengths such as $18-2x$ multiply together to give an area or volume. The recipe is: name each dimension in terms of $x$, multiply, expand, and state which values of $x$ make sense.
A packaging company designs rectangular boxes without a lid from a cardboard sheet of $30 \text{ cm}$ by $20 \text{ cm}$. To do this, they cut out squares of side $x$ at…
A packaging company designs rectangular boxes without a lid from a cardboard sheet of $30 \text{ cm}$ by $20 \text{ cm}$. To do this, they cut out squares of side $x$ at each corner and fold the edges upward. Express the volume $V(x)$ of the box as a polynomial in standard form (expanded). Then, calculate the exact volume if squares of $x = 4 \text{ cm}$ are cut out.
In a 3D design project, we start with a solid cube of edge $x+2 \text{ cm}$. Its interior is hollowed out by extracting another concentric cube of edge $x-1 \text{ cm}$.…
In a 3D design project, we start with a solid cube of edge $x+2 \text{ cm}$. Its interior is hollowed out by extracting another concentric cube of edge $x-1 \text{ cm}$. Express the volume of the remaining material as a polynomial in standard form (expanding the cubed binomials and simplifying). Then, find the value of $x$ if it is known that the volume of the remaining material is exactly $117 \text{ cm}^3$.
Finding unknown coefficients
Often a model polynomial has unknown coefficients. Each fact you are given becomes an equation:
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"The value at $x=c$ is $v$" means $P(c)=v$.
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"The remainder when dividing by $x-c$ is $v$" means $P(c)=v$ (Remainder Theorem).
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"$P$ is divisible by $x-c$" means $P(c)=0$ (Factor Theorem).
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"$P$ has zeros $r_1, r_2, \dots$" means $P(x)=k(x-r_1)(x-r_2)\cdots$ for some constant $k$.
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"$P$ is divisible by $(x-c)^2$" means $P(x)=(x-c)^2\,q(x)$ for some polynomial $q$.
You need as many independent conditions as unknowns.
You can also solve the last example as a system: $P(1)=0$, the quotient from synthetic division by $x-1$ must also be $0$ at $x=1$, and $P(2)=3$. Both methods give the same answer, and the factored form is usually shorter.
The daily profits of a company, in thousands of dollars, are modeled by the polynomial $B(x) = -x^3 + ax^2 + bx + 4$, where $x$ is the day of the week (from 1 to 7). It…
The daily profits of a company, in thousands of dollars, are modeled by the polynomial $B(x) = -x^3 + ax^2 + bx + 4$, where $x$ is the day of the week (from 1 to 7). It is known that on day 1 the profit is zero. Furthermore, when dividing $B(x)$ by $x-2$, the remainder is $2$. Calculate the parameters $a$ and $b$, and factorize the polynomial to determine on which other day of the week the profit is exactly zero again.
The thermal expansion curve of a new polymer is approximated by a cubic polynomial $E(t) = at^3 + bt^2 + ct + d$. In laboratory tests, it has been verified that at…
The thermal expansion curve of a new polymer is approximated by a cubic polynomial $E(t) = at^3 + bt^2 + ct + d$. In laboratory tests, it has been verified that at temperatures $t = 1 \ ^\circ\text{C}$, $t = 2 \ ^\circ\text{C}$ and $t = -1 \ ^\circ\text{C}$, the expansion is zero. However, at $t = 0 \ ^\circ\text{C}$, the expansion is $2 \text{ mm}$. Determine the value of the coefficients $a, b, c, d$ of the polynomial.
The bending curve of a beam subjected to a load is modeled by the polynomial $y(x) = x^4 + ax^3 + bx^2 + cx + d$. It is known that the polynomial is exactly divisible by…
The bending curve of a beam subjected to a load is modeled by the polynomial $y(x) = x^4 + ax^3 + bx^2 + cx + d$. It is known that the polynomial is exactly divisible by $(x-1)^2$. Furthermore, when dividing it by $x+1$ the remainder is $-8$, and when dividing it by $x-2$ the remainder is $4$. Set up the corresponding system of equations and find the values of $a, b, c$ and $d$.