In 4th ESO you already worked with polynomials: adding them, multiplying them and dividing them using Ruffini's rule. In 1st Bachillerato these tools become more general and more useful: they are used to model a company's profit, the path of a drone or the resistance of a circuit, and to solve high-degree equations that would otherwise be impossible to tackle. This section reviews the common ground and signals, in each part, where each idea is developed further.
A polynomial in $x$ is a sum of terms $a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$, where the $a_i$ are real numbers (the coefficients) and the exponents are natural numbers. The degree is the highest exponent with a non-zero coefficient; $a_n$ is the leading coefficient and $a_0$ is the constant term.
To add or subtract polynomials, you collect like terms (terms of the same degree). To multiply, you apply the distributive property: each term of the first polynomial times each term of the second, and then collect like terms.
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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.
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 always has a lower degree than the divisor (or is zero). The algorithm is the same as long division of numbers: you divide the highest-degree term, multiply and subtract, and repeat with what is left.
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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 $R(x)$:
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?
A shortcut when the divisor is $x-a$: Ruffini's rule
If the divisor has degree $1$ and is of the form $x-a$, there is no need to set up the long division: Ruffini's rule works only with the coefficients and is much quicker. The next section develops it in depth; here it is enough to recognise when it can be used.
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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.
Before dividing, it is worth knowing whether it is worthwhile: the remainder theorem says that the remainder of dividing $P(x)$ by $(x-a)$ is, directly, $P(a)$. And 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. Both are extended in the sections "Remainder theorem" and "Factorisation".
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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.
Back to the familiar: common factor, identities and fractions
Factorising starts just as it did in 4th ESO: common factor first, then the standard identities. And an algebraic fraction is simplified in the same way as a numerical fraction, by cancelling the factors that the numerator and denominator have in common.
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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 $P(x) = 3x^3 - 12x$.
Many situations — profits, volumes, signals — are described by a polynomial with an unknown parameter. The conditions of the problem (a known value, an exact remainder) are translated into equations that allow us to find that parameter.
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The daily profits of a company, in thousands of euros, are modelled 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 euros, are modelled 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 factorise the polynomial to determine on which other day of the week the profit is exactly zero again.
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.
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.