\begin{align*} g(x) &= x^{3} + 4x^{2} + 11x - 5 \\ h(x) &= x - 1 \end{align*}
5.3 Remainder theorem
Previous
5.2 Cubic polynomials
|
Next
5.4 Factor theorem
|
5.3 Remainder theorem (EMCGV)
Remainder theorem
Given the following functions:
- \(f(x) = x^{3} + 3x^{2} + 4x + 12\)
- \(k(x) = x - 1\)
- \(g(x) = 4x^{3} -2x^{2} + x -7\)
- \(h(x) = x + 2\)
- Determine \(\dfrac{f(x)}{k(x)}\) and \(\dfrac{g(x)}{h(x)}\).
- Write your answers in the general form: \(a(x)=b(x).Q(x) + R(x)\).
- Determine \(f(1)\) and \(g(-2)\).
- What do you notice?
- Consider the degree of the quotient and the remainder - is there a rule?
- What conclusions can you draw?
- Write a mathematical equation to describe your conclusions.
- Complete the following sentence: a cubic function divided by a linear polynomial gives a quotient with a degree of \(\ldots \ldots\) and a remainder with a degree of \(\ldots \ldots\), which is called a constant.
The Remainder theorem
A polynomial \(p(x)\) divided by \(cx - d\) gives a remainder of \(p\left(\dfrac{d}{c}\right)\).
In words: the value of the remainder \(R\) is obtained by substituting \(x = \frac{d}{c}\) into the polynomial \(p(x)\).
\[R = p \left( \dfrac{d}{c} \right)\]NOTE: PROOF NOT FOR EXAMS
Let the quotient be \(Q(x)\) and let the remainder be \(R\). Therefore we can write:
\begin{align*} p(x) &= (cx - d) \cdot Q(x) + R \\ \therefore p \left( \dfrac{d}{c} \right) &= \left[c\left( \dfrac{d}{c} \right) - d\right] \cdot Q \left( \dfrac{d}{c} \right) + R \\ &= (d - d)\cdot Q \left( \dfrac{d}{c} \right) + R \\ &= 0 \cdot Q \left( \dfrac{d}{c} \right) + R \\ &= R\\ \therefore p \left( \dfrac{d}{c} \right) &= R \end{align*}Worked example 7: Finding the remainder
Use the remainder theorem to determine the remainder when \(p(x) = 3x^{3} + 5x^{2} - x + 1\) is divided by the following linear polynomials:
- \(x + 2\)
- \(2x -1\)
- \(x + m\)
Determine the remainder for each linear divisor
The remainder theorem states that any polynomial \(p(x)\) that is divided by \(cx - d\) gives a remainder of \(p\left(\dfrac{d}{c}\right)\):
- \begin{align*} p(x) &= 3x^{3} + 5x^{2} - x + 1 \\ p(-2) &= 3 \left( -2 \right)^{3} + 5\left( -2 \right)^{2} - \left( -2 \right) + 1 \\ &= 3 \left( -8 \right) + 5\left( 4 \right) +2 + 1 \\ &= -24 + 20 + 3 \\ \therefore R &= -1 \end{align*}
- \begin{align*} p(x) &= 3x^{3} + 5x^{2} - x + 1 \\ p \left( \frac{1}{2} \right) &= 3\left( \frac{1}{2} \right)^{3} + 5\left( \frac{1}{2} \right)^{2} - \left( \frac{1}{2} \right)+ 1 \\ &= 3\left( \frac{1}{8} \right) + 5\left( \frac{1}{4}\right) - \left( \frac{1}{2}\right) + 1 \\ &= \frac{3}{8} + \frac{5}{4} + \frac{1}{2} \\ &= \frac{3}{8} + \frac{10}{8} + \frac{4}{8} \\ \therefore R &= \frac{17}{8} \end{align*}
- \begin{align*} p(x) &= 3x^{3} + 5x^{2} - x + 1 \\ p(m) &= 3\left( -m \right)^{3} + 5\left( -m \right)^{2} - \left( -m \right) + 1 \\ \therefore R &= -3m^{3} + 5m^{2} + m + 1 \end{align*}
Worked example 8: Using the remainder to solve for an unknown variable
Given that \(f(x) = 2x^{3} + x^{2} + kx + 5\) divided by \(2x - 3\) gives a remainder of \(9\frac{1}{2}\), use the remainder theorem to determine the value of \(k\).
Use the remainder theorem to determine the unknown variable \(k\)
From the remainder theorem we know that \(f\left(\frac{3}{2}\right) = 9\frac{1}{2}\) and we can therefore solve for \(k\):
\begin{align*} f(x) &= 2x^{3} + x^{2} + kx + 5 \\ f\left( \frac{3}{2} \right) &= 2 \left( \frac{3}{2} \right)^{3} + \left( \frac{3}{2} \right)^{2} + k\left( \frac{3}{2} \right) + 5 \\ 9\frac{1}{2} &= 2 \left( \frac{27}{8} \right) + \left( \frac{9}{4} \right) + k\left( \frac{3}{2} \right) + 5 \\ 9\frac{1}{2} &= \frac{27}{4} + \frac{9}{4} + \frac{3k}{2} + 5 \\ 9\frac{1}{2} &= \frac{36}{4} + \frac{3k}{2} + 5\\ \therefore 9\frac{1}{2} - 9 - 5 &= \frac{3k}{2} \\ - 4\frac{1}{2} &= \frac{3k}{2} \\ - \frac{9}{2} \times \frac{2}{3} &= k \\ \therefore -3 &= k \end{align*}Write the final answer
Therefore \(k = -3\) and \(f(x) = 2x^{3} + x^{2} - 3x + 5\).
Remainder theorem
Use the remainder theorem to determine the remainder \(R\) when \(g(x)\) is divided by \(h(x)\):
\begin{align*} g(x) &= 2x^{3} - 5x^{2} + 8 \\ h(x) &= 2x - 1 \end{align*}
\begin{align*} g(x) &= 4x^{3} + 5x^{2} + 6x - 1 \\ h(x) &= x + 2 \end{align*}
\begin{align*} g(x) &= -5x^{3} - x^{2} -10x + 9 \\ h(x) &= 5x + 1 \end{align*}
\begin{align*} g(x) &= x^{4} + 5x^{2} + 2x - 8 \\ h(x) &= x + 1 \end{align*}
\begin{align*} g(x) &= 3x^{5} - 8x^{4} + x^{2} + 2 \\ h(x) &= 2 - x \end{align*}
\begin{align*} g(x) &= 2x^{100} - x - 1 \\ h(x) &= x + 1 \end{align*}
Determine the value of \(t\) if \(x^{3} + tx^{2} + 8x + 21\) divided by \(x + 1\) gives a remainder of \(\text{16}\).
Calculate the value of \(m\) if \(2x^{3} - 7x^{2} + mx - 26\) is divided by \(x - 2\) and gives a remainder of \(-\text{24}\).
If \(x^{5} - 2x^{3} - kx - 1\) is divided by \(x - 1\) and the remainder is \(-\frac{1}{2}\), find the value of \(k\) .
Determine the value of \(p\) if \(18x^{3} + px^{2} - 8x + 9\) is divided by \(2x - 1\) and gives a remainder of \(\text{6}\).
If \(x^{3} + x^{2} - x + b\) is divided by \(x - 2\) and the remainder is \(2\frac{1}{2}\), calculate the value of \(b\).
Calculate the value of \(h\) if \(3x^{5} + hx^{4} + 10x^{2} - 21x + 12\) is divided by \(x - 2\) and gives a remainder of \(\text{10}\).
If \(x^{3} + 8x^{2} + mx - 5\) is divided by \(x + 1\) and the remainder is \(n\), express \(m\) in terms of \(n\).
When the polynomial \(2x^{3} + px^{2} + qx + 1\) is divided by \(x + 1\) or \(x - 4\), the remainder is \(\text{5}\). Determine the values of \(p\) and \(q\).
Previous
5.2 Cubic polynomials
|
Table of Contents |
Next
5.4 Factor theorem
|