Polynomials
A company in Guadalajara models its daily profits with a polynomial, and it knows that on day 1 it made no profit. Can you find out on which other day the profit is zero again without trying every single day? With the tools in this section, you can.
Definition, degree, and basic operations
A polynomial in $x$ is a sum of terms $a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$, with real coefficients and natural-number exponents. The degree is the greatest exponent with a nonzero coefficient; $a_n$ is the leading coefficient and $a_0$ the constant term.
Adding or subtracting means combining like terms. Multiplying means applying the distributive property: each term of the first polynomial times each term of the second, and then combining like terms.
Let $P(x) = 2x^3 - 5x^2 + 3x - 1$ and $Q(x) = -x^3 + 4x^2 - 2x + 5$ be polynomials. Calculate the polynomial resulting from the operation $2P(x) - Q(x)$ and express the…
Let $P(x) = 2x^3 - 5x^2 + 3x - 1$ and $Q(x) = -x^3 + 4x^2 - 2x + 5$ be polynomials. Calculate the polynomial resulting from the operation $2P(x) - Q(x)$ and express the result ordered in descending order of degree.
A packaging company designs open-topped rectangular boxes from a sheet of cardboard measuring $30 \text{ cm}$ by $20 \text{ cm}$. To do this, they cut squares of side…
A packaging company designs open-topped rectangular boxes from a sheet of cardboard measuring $30 \text{ cm}$ by $20 \text{ cm}$. To do this, they cut squares of side $x$ from each corner and fold up the edges. Express the volume $V(x)$ of the box as a polynomial in standard expanded form. Then, calculate the exact volume if squares of $x = 4 \text{ cm}$ are cut out.
Polynomial division
Dividing $D(x)$ by $d(x)$ gives a quotient $C(x)$ and a remainder $R(x)$ with $D(x)=d(x)\cdot C(x)+R(x)$, where the remainder has lower degree than the divisor (or is zero). The algorithm is like long division of numbers: you divide the highest-degree term, multiply, subtract, and repeat.
Perform the following polynomial division and clearly indicate the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$:…
Perform the following polynomial division and clearly indicate 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 this remainder?
Synthetic division (Ruffini's rule)
If the divisor has the form $x-a$, you don't need long division: synthetic division (or Ruffini's rule) works only with the coefficients and the number $a$.
Use Ruffini's rule to divide the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by the binomial $x + 2$. State the quotient and the remainder.
Use Ruffini's rule to divide the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by the binomial $x + 2$. State the quotient and the remainder.
The remainder and factor theorems
Before dividing, it helps to know whether it's worth it. The remainder theorem says that the remainder of dividing $P(x)$ by $x-a$ is simply $P(a)$. The factor theorem says that $x-a$ is a factor of $P(x)$ exactly when $P(a)=0$, that is, when $a$ is a root.
Why? From $P(x)=(x-a)C(x)+R$ with $R$ constant, substituting $x=a$ makes the first term vanish and leaves $P(a)=R$.
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 performing the full…
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 performing the full division.
Using the Factor Theorem, check whether 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 your…
Using the Factor Theorem, check whether 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 your calculations.
Factoring and algebraic fractions
Factoring follows an order: first the common factor, then special products (difference of squares, perfect square trinomial, difference of cubes). An algebraic fraction is simplified like a numerical one: you cancel common factors.
Two products worth remembering: $x^3-a^3=(x-a)(x^2+ax+a^2)$ and $x^3+a^3=(x+a)(x^2-ax+a^2)$.
Factorise out the common factor and use algebraic identities to fully factorise the polynomial $P(x) = 3x^3 - 12x$.
Factorise out the common factor and use algebraic identities to fully factorise the polynomial $P(x) = 3x^3 - 12x$.
Simplify the following algebraic fraction as far as possible, factorising the numerator and the denominator first: $\frac{x^2 - 9}{x^2 + 6x + 9}$
Simplify the following algebraic fraction as far as possible, factorising the numerator and the denominator first:
$\frac{x^2 - 9}{x^2 + 6x + 9}$
Fully factorise the fourth-degree polynomial $P(x) = x^4 - 5x^2 + 4$. From its factorised form, state all its real roots.
Fully factorise the fourth-degree polynomial $P(x) = x^4 - 5x^2 + 4$. From its factorised form, state all its real roots.
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 factoring the…
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 factoring the numerator (difference of two cubes) and the denominator first. Then, evaluate the area when $x = 3$ meters.
Operations with algebraic fractions
They are operated on like numerical fractions. To multiply, factor and simplify. To add or subtract, factor the denominators, take the least common multiple (each factor with its greatest exponent), and rewrite everything over that denominator. To divide, multiply by the reciprocal.
For a complex fraction (fractions inside fractions), convert each part into a single factored fraction and divide by multiplying by the reciprocal.
Perform the following multiplication of algebraic fractions and simplify the result: $\frac{2x}{x-1} \times \frac{x^2 - 1}{4x^2}$
Perform the following multiplication of algebraic fractions and simplify the result:
$\frac{2x}{x-1} \times \frac{x^2 - 1}{4x^2}$
In an electrical circuit, the equivalent resistance $R_{eq}$ is given by the following algebraic fraction expression:…
In an electrical circuit, the equivalent resistance $R_{eq}$ is given by the following algebraic fraction expression:
$R_{eq}(x) = \frac{2}{x^2-1} - \frac{1}{x-1} + \frac{x}{x+1}$
Perform the indicated operations, calculate the lowest common multiple of the denominators, and simplify the resulting fraction as far as possible. What is the value of the resistance if $x = 3 \ \Omega$?
In the design of a gear mechanism, the gear ratio is modelled by the following complex algebraic expression:…
In the design of a gear mechanism, the gear ratio is modelled 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 far as possible by factorising all numerators and denominators first. What mathematical (and therefore physical) problem occurs with the gear 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:
$S(x) = \frac{x^4 - 2x^3 - 7x^2 + 8x + 12}{x^3 - 5x^2 + 2x + 8}$
Fully factorise the numerator and the denominator using Ruffini's rule and by solving quadratic equations. Simplify the fraction as far as possible and determine, if it exists, for which real value of $x$ the simplified fraction is equal to $3$.
The binomial theorem
For $(a+b)^n$ the coefficients come from Pascal's triangle (Tartaglia's triangle): each number is the sum of the two above it.
Expand the power of the binomial $(2x - y)^4$ fully, using the Binomial Theorem or Pascal's triangle.
Expand the power of the binomial $(2x - y)^4$ fully, using the Binomial Theorem or Pascal's triangle.
In a study of genetic combinations, a certain probability distribution is related to the expansion of the binomial $\left(x^2 - \frac{1}{x}\right)^6$. Find the exact…
In a study of genetic combinations, a certain probability distribution 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 independent of $x$) 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 this expansion.
Polynomials that model real problems
Profits, volumes, deformations: many situations are described by a polynomial with unknown parameters. Each condition (a known value, a remainder, divisibility by $x-a$, $a$ being a root) translates into an equation; with as many independent equations as parameters, the system can be solved.
Back to the beginning: the already-known factor $(x-1)$ and synthetic division gave us the other day of zero profit without trying them one by one.
The daily profits of a company, in thousands of pesos, 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…
The daily profits of a company, in thousands of pesos, 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 factor 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 was found that at temperatures…
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 was found 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 deflection curve of a beam under load is modelled by the polynomial $y(x) = x^4 + ax^3 + bx^2 + cx + d$. It is known that the polynomial is exactly divisible by…
The deflection curve of a beam under load is modelled 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 divided by $x+1$ the remainder is $-8$, and when divided by $x-2$ the remainder is $4$. Set up the corresponding system of equations and find the values of $a, b, c$ and $d$.
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 factorise the resulting polynomial to find all possible semi-heights $x$ that satisfy the geometric condition.
Using the Factor Theorem, prove that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n \ge 1$. Then, apply this property to find…
Using the Factor Theorem, prove that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n \ge 1$. Then, apply this property to find the obvious root and fully factorise the polynomial $P(x) = x^5 - 32$ over the set of real numbers.
In a 3D design project, a solid cube with edge length $x+2 \text{ cm}$ is used as a starting point. Its interior is hollowed out by removing another concentric cube of…
In a 3D design project, a solid cube with edge length $x+2 \text{ cm}$ is used as a starting point. Its interior is hollowed out by removing another concentric cube of edge length $x-1 \text{ cm}$. Express the volume of 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 remaining material is exactly $117 \text{ cm}^3$.