(1) Find the value of the polynomial \(2x - 2x{^3} + 7\) using given values for \(x\).
(i) \(x = 3\)
Solution:

Let, \(p(x) = 2x - 2x{^3} + 7\)
∴ \(p(3) = 2 \times 3 - 2 \times 3{^3} + 7\)
∴ \(p(3) = 6 - 54 + 7\)
∴ \(p(3) = -\:41\)

(1) Find the value of the polynomial \(2x - 2x{^3} + 7\) using given values for \(x\).
(ii) \(x = -\:1\)
Solution:

Let, \(p(x) = 2x - 2x{^3} + 7\)
∴ \(p(-\:1) = 2 \times (-\:1) - 2 \times (-\:1){^3} + 7\)
∴ \(p(-\:1) = -\:\cancel{2} + \cancel{2} + 7\)
∴ \(p(-\:1) = 7\)

(1) Find the value of the polynomial \(2x - 2x{^3} + 7\) using given values for \(x\).
(iii) \(x = 0\)
Solution:

Let, \(p(x) = 2x - 2x{^3} + 7\)
∴ \(p(0) = 2 \times 0 - 2 \times (0){^3} + 7\)
∴ \(p(0) = 0 -\: 0 + 7\)
∴ \(p(0) = 7\)

(2) For each of the following polynomials, find \(p(1)\), \(p(0)\) and \(p(-\:2)\).
(i) \(p(x) = x^{3}\)
Solution:

 \(p(x) = x^{3}\)
∴ \(p(1) = (1)^{3}\)
∴ \(p(1) = 1\)

 \(p(x) = x^{3}\)
∴ \(p(0) = (0)^{3}\)
∴ \(p(0) = 0\)

 \(p(x) = x^{3}\)
∴ \(p(-\:2) = (-\:2)^{3}\)
∴ \(p(-\:2) = -\:8\)

(2) For each of the following polynomials, find \(p(1)\), \(p(0)\) and \(p(-\:2)\).
(ii) \(p(y) = y^{2} - 2y + 5\)
Solution:

 \(p(y) = y^{2} - 2y + 5\)
∴ \(p(1) = (1)^{2} - 2 \times 1 + 5\)
∴ \(p(1) = 1 - 2 + 5\)
∴ \(p(1) = 4\)

 \(p(y) = y^{2} - 2y + 5\)
∴ \(p(0) = (0)^{2} - 2 \times 0 + 5\)
∴ \(p(0) = 0 - 0 + 5\)
∴ \(p(0) = 5\)

 \(p(y) = y^{2} - 2y + 5\)
∴ \(p(-\:2) = (-\:2)^{2} - 2 \times (-2) + 5\)
∴ \(p(-\:2) = 4 + 4 + 5\)
∴ \(p(-\:2) = 13\)

(2) For each of the following polynomials, find \(p(1)\), \(p(0)\) and \(p(-\:2)\).
(iii) \(p(x) = x^{4} - 2x^{2} - x\)
Solution:

 \(p(x) = x^{4} - 2x^{2} - x\)
∴ \(p(1) = (1)^{4} - 2 \times (1)^{2} - 1\)
∴ \(p(1) = \cancel{1} - 2 - \cancel{1}\)
∴ \(p(1) = - 2\)

 \(p(x) = x^{4} - 2x^{2} - x\)
∴ \(p(0) = (0)^{4} - 2 \times (0)^{2} - 0\)
∴ \(p(0) = 0 - 0 - 0\)
∴ \(p(0) = 0\)

 \(p(x) = x^{4} - 2x^{2} - x\)
∴ \(p(-\:2) = (-\:2)^{4} - 2 \times (-\:2)^{2} - (-\:2)\)
∴ \(p(-\:2) = 16 - 8 + 2\)
∴ \(p(-\:2) = 10\)

(3) If the value of the polynomial \(m^{3} + 2m + a\) is 12 for \(m = 2\), then find the value of \(a\).
Solution:

Let, \(p(m) = m^{3} + 2m + a\)
∴ \(p(2) = (2)^{3} + 2 \times (2) + a\)
∴ \(p(2) = 8 + 4 + a\)
∴ \(p(2) = 12 + a\) ... (i)

But, \(p(2) = 12\) ... (Given)
∴ \(12 + a = 12\)
∴ \(a = 12 - 12\)
∴ \(a = 0\)

(4) For the polynomial \(mx^{2} - 2x + 3\), if \(p(- 1) = 7\), then find \(m\).
Solution:

Let, \(p(m) = mx^{2} - 2x + 3\)
∴ \(p(-\:1) = m(-\:1)^{2} - 2 \times (-\:1) + 3\)
∴ \(p(-\:1) = m + 2 + 3\)
∴ \(p(-\:1) = m + 5\) ... (i)

But, \(p(-\:1) = 7\) ... (Given)
∴ \(m + 5 = 7\)
∴ \(m = 7 - 5\)
∴ \(m = 2\)

(5) Divide the first polynomial by the second polynomial and find the remainder using remainder theorem:
(i) \((x^{2} - 7x + 9);\ (x + 1)\)
Solution:

Let, \(p(x) = x^{2} + 7x + 9\)
And, Divisor = \(x + 1\)
Take \(x = -\:1\),
∴ \(p(-\:1) = (-\:1)^{2} + 7 \times (-\:1) + 9\)
∴ \(p(-\:1) = 1 + 7 + 9\)
∴ \(p(-\:1) = 17\)
∴ The remainder is \(17\).

(5) Divide the first polynomial by the second polynomial and find the remainder using remainder theorem:
(ii) \((2x^{3} - 2x^{2} + ax - a);\ (x - a)\)
Solution:

Let, \(p(x) = 2x^{3} - 2x^{2} + ax - a\)
And, Divisor = \(x - a\)
Take \(x = a\),
∴ \(p(a) = 2a^{3} - 2a^{2} + a \times a - a\)
∴ \(p(a) = 2a^{3} \bbox[yellow, 5pt, border: 2px dotted red]{- 2a^{2} + a^{2}} - a\)
∴ \(p(a) = 2a^{3} - a^{2} - a\)
∴ The remainder is \(2a^{3} - a^{2} - a\).

(5) Divide the first polynomial by the second polynomial and find the remainder using remainder theorem:
(iii) \((54m^{3} + 18m^{2} - 27m + 5);\ (m - 3)\)
Solution:

Let, \(p(m) = 54m^{3} + 18m^{2} - 27m + 5\)
And, Divisor = \(m - 3\)
Take \(m = 3\),
∴ \(p(3) = 54 \times (3)^{3} + 18 \times (3)^{2} - 27 \times 3 + 5\)
∴ \(p(3) = 54 \times 27 + 18 \times 9 - 81 + 5\)
∴ \(p(3) = 1458 + 162 - 81 + 5\)
∴ \(p(3) = 1544\)
∴ The remainder is \(1544\).

(6) If the polynomial \(y^{3} - 5y^{2} + 7y + m\) is divided by \(y + 2\), and the remainder is 50, then find the value of \(m\).
Solution:

Let, \(p(y) = y^{3} - 5y^{2} + 7y + m\)
Divisor = \(y + 2\)
Take \(y = -\:2\),
∴ \(p(-\:2) = (-\:2)^{3} - 5(-\:2)^{2} + 7 \times (-\:2) + m\)
∴ \(p(-\:2) = -\:8 - 5 \times 4 -\: 14 + m\)
∴ \(p(-\:2) = -\:8 - 20 -\: 14 + m\)
∴ \(p(-\:2) = -\:42 + m\)
∴ Remainder \(= -\:42 + m\) ... (i)

But, Remainder \(= 50\) ... (Given)

∴ \(-\:42 + m = 50\)
∴ \(m = 50 + 42\)
∴ \(m = 92\)

The value of \(m\) is \(92\).

(7) Use the factor theorem to determine whether \(x - 3\)
is a factor of \(x²\ +\ 2x\ -\ 3\) or not.
Solution:

Let, \(p(x) = x²\ +\ 2x\ -\ 3\)
∴ \(p(-\:3) = (-\:3)^{2} + 2(-\:3)-\:3\)
∴ \(p(-\:3) = 9 - 6 -\: 3\)
∴ \(p(-\:3) = 0\)
∴ \(x + 3\) is a factor of \(\:x²\:+\:2x\:-\:3\)

(8) If \((x - 2)\) is a factor of \(x^{3} - mx^{2} + 10x - 20\), then find the value of \(m\).
Solution:

Here, we will use the factor theorem:
\((x - 2)\) is a factor of \(x^{3} - mx^{2} + 10x - 20\) ... (Given)
∴ By factor theorem,
 \(p(2) = 0\)
∴ \((2)^{3} - m \times (2)^{2} + 10 \times 2 - 20 = 0\)
∴ \(8 - 4m + \cancel{20} - \cancel{20} = 0\)
∴ \(8 - 4m = 0\)
∴ \(- 4m = - 8\)
i.e. \(4m = 8\)

∴ \(\displaystyle m = \frac{\cancelto{2}{8}}{\cancelto{1}{4}}\)

∴ \(m = 2\)

(9) By using factor theorem, in the following examples, determine whether \(q(x)\) is a factor of \(p(x)\) or not:
(i) \(p(x) = x^{3} - x^{2} - x - 1\), \(q(x) = x - 1\)
Solution:

\(p(x) = x^{3} - x^{2} - x - 1\)
Take \(x = 1\),
∴ \(p(1) = (1)^{3} - (1)^{2} - 1 - 1\)
∴ \(p(1) = \cancel{1} - 1 - 1 - \cancel{1}\)
∴ \(p(1) = - 2\)
∴ \(p(1) \ne 0\)
∴ \(q(x)\) is not a factor of \(p(x)\).

(9) By using factor theorem, in the following examples, determine whether \(q(x)\) is a factor of \(p(x)\) or not:
(ii) \(p(x) = 2x^{3} - x^{2} - 45\), \(q(x) = x - 3\)
Solution:

\(p(x) = 2x^{3} - x^{2} - 45\)
Take \(x = 3\),
∴ \(p(3) = 2(3)^{3} - (3)^{2} - 45\)
∴ \(p(3) = 54 - 9 - 45\)
∴ \(p(3) = 0\)
∴ \(q(x)\) is a factor of \(p(x)\).

(10) If \((x^{31} + 31)\) is divided by \((x + 1)\), then find the remainder.
Solution:

Let, \(p(x) = x^{31} + 31\)
Take \(x =\:-\:1\),
∴ \(p(-\:1) = (-\:1)^{31} + 31\)
∴ \(p(-\:1) = -\:1 + 31\)
∴ \(p(-\:1) = 30\)
∴ The remainder is \(30\).

(11) Show that \(m - 1\) is a factor of \(m^{21} - 1\) and \(m^{22} - 1\).
Solution:

Let, \(p(m) = m^{21} - 1\)
Take \(m = 1\),
∴ \(p(1) = (1)^{21} - 1\)
∴ \(p(1) = 1 - 1\)
∴ \(p(1) = 0\)
∴ \(m - 1\) is a factor of \(m^{21} - 1\).

Let, \(q(m) = m^{22} - 1\)
Take \(m = 1\),
∴ \(q(1) = (1)^{22} - 1\)
∴ \(q(1) = 1 - 1\)
∴ \(q(1) = 0\)
∴ \(m - 1\) is a factor of \(m^{22} - 1\).

(12) If \(\displaystyle x - 2\) and \(\displaystyle x - \frac{1}{2}\) both are the factors of the polynomial \(nx^{2} - 5x + m\), then show that \(m = n = 2\).
Solution:

Let, \(p(x) = nx^{2} - 5x + m\)
Now, \(x - 2\) is a factor of \(p(x)\).
∴ \(p(2) = 0\)
∴ \(n(2)^{2} - 5(2) + m = 0\)
∴ \(4n - 10 + m = 0\)
∴ \(4n + m = 10\) ... (i)

Also, \(\displaystyle x - \frac{1}{2}\) is a factor of \(p(x)\).

∴ \(\displaystyle p\left[ \frac{ 1 }{ 2 } \right] = 0\)

∴ \(\displaystyle n\left[ \frac{ 1 }{ 2 } \right]^{2} - 5\left[ \frac{ 1 }{ 2 } \right] + m = 0\)

∴ \(\displaystyle n \times\frac{1}{4} - 5 \times \frac{1}{2} + m = 0\)

∴ \(\displaystyle \frac{n}{4} - \frac{5}{2} + m = 0\)

Multiplying both sides by 4,

 \(\displaystyle \frac{n}{\cancel {4}} \times \cancel {4} - \frac{5}{\cancel {2}} \times \cancelto{2}{4} + m \times 4 = 0 \times 4\)

∴ \(n - 10 + 4m = 0\)
∴ \(n + 4m = 10\) ... (ii)

From (i) and (ii),
 \(4n + m = n + 4m\)
∴ \(4n - n = 4m - m\)
∴ \(\cancel{3}n = \cancel{3}m\)
∴ \(n = m\) ... (iii)

Substituting this value of \(n\) in equation (i),
 \(4n + m = 10\) ... (i)
∴ \(4m + m = 10\)
∴ \(5m = 10\)

∴ \(\displaystyle m = \frac{\cancelto{5}{10}}{\cancel{2}}\)

∴ \(m = 2\) ... (iv)

From (iii) and (iv),
∴ \(m = n = 2\)

(13) (i) If If \(p(x) = 2 + 5x\), then find \(p(2) + p(\:-\: 2)\:-\:p(1)\).
Solution:

 \(p(x) = 2 + 5x\) ... (Given)
∴ \(p(2) = 2 + 5 \times 2\)
∴ \(p(2) = 2 + 10\)
∴ \(p(2) = 12\) ... (i)

Also, \(p(x) = 2 + 5x\) ... (Given)
∴ \(p(\:-\:2) = 2 + 5 \times \:-\:2\)
∴ \(p(\:-\:2) = 2 \:-\:10\)
∴ \(p(\:-\:2) = \:-\:8\) ... (ii)

 \(p(x) = 2 + 5x\) ... (Given)
∴ \(p(1) = 2 + 5 \times 1\)
∴ \(p(1) = 2 + 5\)
∴ \(p(2) = 7\) ... (iii)

Adding (i), (ii) and (iii),
 \(p(2) + p(\:-\:2) \:-\: p(1) = 12 \:-\:8 \:-\: 7\)
∴ \(p(2) + p(\:-\:2) \:-\: p(1) = \:-\: 3\)

(13) (ii) If \(p(x) = 2x^{2} - 5\sqrt{3}x + 5\), then find \(p(5\sqrt{3})\).
Solution:

  \(p(x) = 2x^{2} - 5\sqrt{3}x + 5\) ... (Given)
∴ \(p(5\sqrt{3}) = 2 \times (5\sqrt{3})^{2} - 5\sqrt{3} \times 5\sqrt{3} + 5\)
∴ \(p(5\sqrt{3}) = 2 \times 25 \times 3 - 25 \times 3 + 5\)
∴ \(p(5\sqrt{3}) = 150 - 75 + 5\)
∴ \(p(5\sqrt{3}) = 80\)

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