Polynomials
A box factory in Santiago receives cardboard sheets that measure $30$ cm by $20$ cm. To make an open-top box, it cuts a square of side $x$ out of each corner and folds the edges up. What expression gives the volume of the box depending on the value of $x$? That expression is a polynomial, and in this section you review the tools from Secondary Year 3 for working with them: operating, dividing, and factoring. With them you will solve high-degree equations in the following sections.
Definition, degree, and 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 $a_i$ and natural-number exponents. The degree is the highest exponent with a nonzero coefficient. $a_n$ is the leading coefficient and $a_0$ is the constant term.
Adding or subtracting polynomials means combining like terms (those of the same degree). Multiplying means applying the distributive property: each term of the first times each term of the second, and then combining.
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.
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 $117 \text{ cm}^3$.
Division of polynomials
Dividing $D(x)$ by $d(x)$ gives a quotient $C(x)$ and a remainder $R(x)$ such that
$$D(x)=d(x)\cdot C(x)+R(x),$$with the remainder of lower degree than the divisor (or zero). The algorithm is the long division of numbers: you divide the leading terms, multiply, subtract, and repeat with what is left.
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)
When the divisor has the form $x-a$, synthetic division (Ruffini's rule) is a shortcut: you work only with the coefficients and the number $a$. To divide by $x+2$ you use $a=-2$, because $x+2=x-(-2)$.
Steps: write the coefficients of $P(x)$ in order (with $0$ where a term is missing). Bring down the first one. Multiply it by $a$, add it to the next coefficient, and repeat. The last number is the remainder, and the previous ones are the coefficients of the quotient, which has one degree less.
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.
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.
The Remainder and Factor Theorems
Before dividing, it helps to know whether it is worth it. The Remainder Theorem says that the remainder of dividing $P(x)$ by $x-a$ is $P(a)$.
Why? By the division, $P(x)=(x-a)C(x)+R$, with $R$ constant (the remainder has degree less than 1). If you substitute $x=a$, the first term vanishes and you get $P(a)=R$.
The Factor Theorem is a direct consequence: $x-a$ is a factor of $P(x)$ exactly when the remainder is zero, that is, when $P(a)=0$ (when $a$ is a root).
These theorems are also useful for finding unknown coefficients: each condition in the problem ("it equals $0$ at $x=1$," "the remainder of dividing by $x-2$ is $4$") gives an equation, $P(1)=0$ or $P(2)=4$. If $P(x)$ is divisible by $(x-a)^2$, then $a$ is a root of $P(x)$ and also of the quotient of dividing by $x-a$: that is two conditions.
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.
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 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 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 $d$.
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.
Factoring polynomials
Factoring means writing the polynomial as a product of polynomials of lower degree. Follow this order:
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Common factor.
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Special products: $a^2-b^2=(a-b)(a+b)$, $a^2\pm 2ab+b^2=(a\pm b)^2$, and the sum and difference of cubes, $a^3-b^3=(a-b)(a^2+ab+b^2)$ and $a^3+b^3=(a+b)(a^2-ab+b^2)$.
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Quadratic trinomials: find the roots with the quadratic formula.
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Rational zeros and synthetic division: try the divisors of the constant term (divided by those of the leading coefficient), lower the degree with synthetic division, and keep factoring.
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Biquadratics $ax^4+bx^2+c$: substitute $u=x^2$.
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 $P(x) = 3x^3 - 12x$.
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.
Powers of a binomial
The coefficients of $(a+b)^n$ form Pascal's triangle (also called Tartaglia's triangle). Each number is the sum of the two above it:
| $n$ | Coefficients |
|---|---|
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
| 6 | 1 6 15 20 15 6 1 |
The general formula is
$$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k,\qquad \binom{n}{k}=\frac{n!}{k!\,(n-k)!}.$$The term in position $k+1$ is $T_{k+1}=\binom{n}{k}a^{n-k}b^k$. The exponents of $a$ go down and those of $b$ go up, and their sum is always $n$. For example, $\binom{n}{2}=\dfrac{n(n-1)}{2}$.
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.