If a company's revenue is $I(x) = 300x - 2x^2$ and its cost is $C(x) = 50x + 200$, the profit $B(x) = I(x) - C(x)$ is another polynomial. Operating with polynomials — adding, multiplying and dividing them — is the tool that turns several formulae into a single one, ready to be evaluated or factorised.
To add or subtract polynomials, you collect like terms. The minus sign in a subtraction affects all the terms of the second polynomial.
To multiply, each term of the first polynomial is multiplied by every term of the second; an area rectangle model makes this easy to visualise.
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Given the polynomials $A(x) = 3x^2 - 5x + 1$ and $B(x) = -x^2 + 4x - 6$, calculate the expression resulting from carrying out the operation $2 \cdot A(x) - B(x)$.
Given the polynomials $A(x) = 3x^2 - 5x + 1$ and $B(x) = -x^2 + 4x - 6$, calculate the expression resulting from carrying out the operation $2 \cdot A(x) - B(x)$.
These are three products that occur so often that it is worth recognising them without expanding step by step: they save time when calculating and, in reverse, they are the first tool for factorising.
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Expand the following expressions using the appropriate algebraic identities: a) $(3x + 4)^2$ b) $(2x - 5)^2$
Expand the following expressions using the appropriate algebraic identities:
Dividing $D(x)$ by $d(x)$ gives a quotient $C(x)$ and a remainder $R(x)$ with $D(x) = d(x)\cdot C(x) + R(x)$, just like long division of numbers, and the remainder has a degree lower than the divisor (or is $0$).
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Divide the polynomial $D(x) = 4x^4 - 2x^3 + 6x^2 - x + 5$ by the binomial $d(x) = x^2 + 2$. Clearly state the quotient polynomial $C(x)$ and the remainder polynomial…
Divide the polynomial $D(x) = 4x^4 - 2x^3 + 6x^2 - x + 5$ by the binomial $d(x) = x^2 + 2$. Clearly state the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$.
Check whether the polynomial $P(x) = x^4 - 5x^3 + 7x^2 - 5x + 6$ is divisible by the binomial $(x^2 + 1)$ by carrying out the corresponding division. If so, write the…
Check whether the polynomial $P(x) = x^4 - 5x^3 + 7x^2 - 5x + 6$ is divisible by the binomial $(x^2 + 1)$ by carrying out the corresponding division. If so, write the factorisation of $P(x)$ as the product of two polynomials.
When the divisor has the form $x - a$, there is a much quicker shortcut than general division: Ruffini's rule. You only work with the coefficients of the dividend and the value $a$.
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Use Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by $(x - 2)$. Write down the quotient and remainder obtained.
Use Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by $(x - 2)$. Write down the quotient and remainder obtained.
Knowing that dividing a polynomial $P(x)$ by $(x - 3)$ gives a quotient of $C(x) = 4x^2 - 2x + 1$ and a remainder of $R = -7$, reconstruct the polynomial $P(x)$ and…
Knowing that dividing a polynomial $P(x)$ by $(x - 3)$ gives a quotient of $C(x) = 4x^2 - 2x + 1$ and a remainder of $R = -7$, reconstruct the polynomial $P(x)$ and express it in expanded form.
Calculate the value of the parameter $k$ such that the division of the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$ by the binomial $(x + 2)$ has a remainder equal to $10$.…
Calculate the value of the parameter $k$ such that the division of the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$ by the binomial $(x + 2)$ has a remainder equal to $10$. Use the remainder theorem to justify your answer.
Find the values of the parameters $a$ and $b$ such that the polynomial $P(x) = x^3 + ax^2 + bx - 6$ is exactly divisible by $(x - 2)$ and, at the same time, leaves a…
Find the values of the parameters $a$ and $b$ such that the polynomial $P(x) = x^3 + ax^2 + bx - 6$ is exactly divisible by $(x - 2)$ and, at the same time, leaves a remainder of $-12$ when divided by $(x + 1)$.
Real-life problems chain several of these operations together: build a polynomial from the operations in the statement, simplify it and, if necessary, evaluate it or divide it.
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A rectangular garden has dimensions of $(2x + 5)$ metres long and $(2x - 5)$ metres wide. Inside it, a square fountain with side length $(x - 1)$ metres will be built,…
A rectangular garden has dimensions of $(2x + 5)$ metres long and $(2x - 5)$ metres wide. Inside it, a square fountain with side length $(x - 1)$ metres will be built, and the rest will be covered with lawn. Express the area of the lawn area as a simplified polynomial in $x$.
The production cost of $x$ units of an electronic device is given by the polynomial $C(x) = 50x + 200$. The retail price of each unit depends on the quantity…
The production cost of $x$ units of an electronic device is given by the polynomial $C(x) = 50x + 200$. The retail price of each unit depends on the quantity manufactured and follows the expression $p(x) = 300 - 2x$.
Find the profit polynomial $B(x)$, knowing that profit is equal to total revenue (quantity times price) minus total costs. What will the profit be if $40$ units are manufactured and sold?
The height in metres of a ball thrown vertically as a function of time in seconds is given by the polynomial $H(t) = -5t^2 + 20t + 2$. A second ball, thrown from a…
The height in metres of a ball thrown vertically as a function of time in seconds is given by the polynomial $H(t) = -5t^2 + 20t + 2$. A second ball, thrown from a different point, follows a trajectory described by $G(t) = -5t^2 + 15t + 10$.
Find the polynomial $D(t)$ representing the difference in height between the first and second balls ($D(t) = H(t) - G(t)$). At what instant $t$ are both balls at the same height?