To find the remainder when $P(x)=x^{2024}-3x^{2023}+2x^5-7x+4$ is divided by $x-3$, nobody carries out the whole division: with a single substitution, $P(3)$, the problem is solved in seconds. Imagine this is a code key that a Dublin phone company generates from that polynomial. The Factor Theorem and its remainder form are that substitution, and in Fifth Year Higher Level maths it is used with more confidence: several conditions at once, enormous indices, or divisors of degree $2$. A Leaving Cert question can give you a polynomial with unknown coefficients and ask you to use the Factor Theorem to find them.
The remainder when dividing by (x − a)
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Without performing the division, calculate the remainder when dividing the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by $x - 2$.
Without performing the division, calculate the remainder when dividing the polynomial $P(x) = 3x^4 - 2x^3 + 5x - 7$ by $x - 2$.
Calculate the numerical value of the polynomial $P(x) = -x^5 + 3x^3 - 2x^2 + x - 5$ for $x = -1$ and state the meaning of this value with respect to the polynomial…
Calculate the numerical value of the polynomial $P(x) = -x^5 + 3x^3 - 2x^2 + x - 5$ for $x = -1$ and state the meaning of this value with respect to the polynomial division of $P(x)$ by $x + 1$.
The position of an object moving in a straight line as a function of time $t$ (in seconds) is given by the polynomial $x(t) = -2t^3 + 8t^2 - 5t + 3$ (in metres). Use the…
The position of an object moving in a straight line as a function of time $t$ (in seconds) is given by the polynomial $x(t) = -2t^3 + 8t^2 - 5t + 3$ (in metres). Use the Remainder Theorem to calculate the position of the object at time $t = 3$ and explain the conceptual relationship between the value found and the polynomial division of $x(t)$ by $t - 3$.
If the remainder $P(a)$ turns out to be $0$, the division is exact: $(x-a)$ is a factor of $P(x)$ and $a$ is a root. It is the same idea as the remainder property, applied to the particular case in which the remainder vanishes.
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Determine whether the polynomial $Q(x) = x^3 + 4x^2 - x - 4$ is divisible by $x + 4$ by applying the Remainder Theorem.
Determine whether the polynomial $Q(x) = x^3 + 4x^2 - x - 4$ is divisible by $x + 4$ by applying the Remainder Theorem.
When the polynomial has an unknown coefficient and a particular remainder is known (or required), we substitute in the same way, and the condition becomes an equation in that unknown.
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Find the value of the parameter $k$ so that the remainder of dividing $P(x) = 2x^3 - kx^2 + 3x + 1$ by $x - 1$ is $4$.
Find the value of the parameter $k$ so that the remainder of dividing $P(x) = 2x^3 - kx^2 + 3x + 1$ by $x - 1$ is $4$.
Determine the value of $k$ so that the remainder of dividing the polynomial $P(x) = 2x^3 - kx^2 + x - k$ by $x - 2$ is identical to the remainder of dividing the…
Determine the value of $k$ so that the remainder of dividing the polynomial $P(x) = 2x^3 - kx^2 + x - k$ by $x - 2$ is identical to the remainder of dividing the polynomial $Q(x) = x^2 - kx + 3$ by $x + 1$.
When there are two unknown parameters, two conditions are needed (two remainders, or one remainder and one divisibility condition), and each gives an equation. Together they form a system that is solved like any other.
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Find the values of the parameters $m$ and $n$ so that the polynomial $P(x) = x^3 + mx^2 + nx - 10$ is divisible by $x - 2$ and leaves a remainder of $-24$ when divided…
Find the values of the parameters $m$ and $n$ so that the polynomial $P(x) = x^3 + mx^2 + nx - 10$ is divisible by $x - 2$ and leaves a remainder of $-24$ when divided by $x + 2$.
Calculate the values of $a$ and $b$ knowing that the polynomial $P(x) = ax^4 + bx^3 - x^2 + 2x - 4$ is simultaneously divisible by $(x - 1)$ and by $(x + 1)$.
Calculate the values of $a$ and $b$ knowing that the polynomial $P(x) = ax^4 + bx^3 - x^2 + 2x - 4$ is simultaneously divisible by $(x - 1)$ and by $(x + 1)$.
A cubic polynomial $P(x)$ has leading coefficient $2$. Knowing that it is divisible by $(x - 1)$ and by $(x + 2)$, and that when divided by $(x - 2)$ it leaves a…
A cubic polynomial $P(x)$ has leading coefficient $2$. Knowing that it is divisible by $(x - 1)$ and by $(x + 2)$, and that when divided by $(x - 2)$ it leaves a remainder of $16$, determine the full expression of the polynomial $P(x)$ in expanded form.
Find the expanded expression of a cubic polynomial $P(x)$ such that, when divided by $x+1$, $x-2$, and $x+3$, it leaves the same remainder equal to $5$ in all cases, and…
Find the expanded expression of a cubic polynomial $P(x)$ such that, when divided by $x+1$, $x-2$, and $x+3$, it leaves the same remainder equal to $5$ in all cases, and when divided by $x-1$ it leaves a remainder of $-7$.
Note: Think about the relationship between the auxiliary polynomial $P(x) - 5$ and the Factor Theorem.
Before calculating exactly, substitute $x=a$ and compare the two highest-degree terms. The remainder is $P(a)$, so whichever contribution is larger in size decides its sign. A leading term with a positive coefficient is not always the larger one: for large values of $a$ the highest-degree term eventually dominates, but for moderate values of $a$ a lower-degree term with a big negative coefficient can be larger in size.
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Given the polynomial $P(x) = 3x^4 - 20x^3 + 5x - 12$: a) Before performing any detailed calculation on paper, mentally estimate the sign of the remainder when dividing…
Given the polynomial $P(x) = 3x^4 - 20x^3 + 5x - 12$:
a) Before performing any detailed calculation on paper, mentally estimate the sign of the remainder when dividing $P(x)$ by $x - 4$. To do this, analyze the two highest-degree terms when substituting $x = 4$ and reason whether the positive or negative contribution will dominate.
b) Apply the remainder theorem to calculate the exact value of this remainder. Check whether your qualitative estimate was correct and justify why the leading term ($3x^4$) failed to make the final value positive.
In a cryptographic security system, a key is generated by evaluating a polynomial of very high degree. The key corresponds exactly to the remainder of dividing the…
In a cryptographic security system, a key is generated by evaluating a polynomial of very high degree. The key corresponds exactly to the remainder of dividing the polynomial $P(x) = x^{2024} - 3x^{2023} + 2x^5 - 7x + 4$ by the binomial $x - 3$. Calculate the numerical value of this security key, justifying your answer.
Prove, applying the Remainder Theorem, that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n$. Use this result to deduce, with…
Prove, applying the Remainder Theorem, that the polynomial $P(x) = x^n - a^n$ is always divisible by $x - a$ for any natural number $n$. Use this result to deduce, with reasoning, the remainder of the division of $x^5 - 32$ by $x - 2$.
The remainder when dividing by a polynomial of degree 2
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Find the remainder when dividing the polynomial $P(x) = (x - 2)^{200} + (x - 1)^{100} - 5$ by the quadratic polynomial $Q(x) = (x - 1)(x - 2)$. Note: Remember that the…
Find the remainder when dividing the polynomial $P(x) = (x - 2)^{200} + (x - 1)^{100} - 5$ by the quadratic polynomial $Q(x) = (x - 1)(x - 2)$.
Note: Remember that the remainder when dividing by a polynomial of degree 2 is at most of the form $mx + n$. Apply the Remainder Theorem by strategically evaluating at the roots of the divisor.
Find the expanded expression of a cubic polynomial $P(x)$ such that, when divided by $x+1$, $x-2$, and $x+3$, it leaves the same remainder equal to $5$ in all cases, and…
Find the expanded expression of a cubic polynomial $P(x)$ such that, when divided by $x+1$, $x-2$, and $x+3$, it leaves the same remainder equal to $5$ in all cases, and when divided by $x-1$ it leaves a remainder of $-7$.
Note: Think about the relationship between the auxiliary polynomial $P(x) - 5$ and the Factor Theorem.