3 Marks Question

Question 13 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = 2x³ + 3x² - 11x - 3, $\text{g}(\text{x})=\Big(\text{x}+\frac{1}{2}\Big).$

Answer

p(x) =2x³ + 3x² - 11x - 3

$\text{g}(\text{x})=\Big(\text{x}+\frac{1}{2}\Big)=\Big[\text{x}-\Big(-\frac{1}{2}\Big)\Big]$

By remainder theorem, when p(x) is divided by $\Big(\text{x}+\frac{1}{2}\Big),$ then the remainder $=\text{p}\Big(-\frac{1}{2}\Big).$

Putting $\text{x}=-\frac{1}{2}$ in p(x), we get

$\text{p}\Big(-\frac{1}{2}\Big)=2\times\Big(-\frac{1}{2}\Big)^3\\+3\times\Big(-\frac{1}{2}\Big)^2-11\times\Big(-\frac{1}{2}\Big)-3$

$=-\frac{1}{4}+\frac{3}{4}+\frac{11}{2}-3$

$=\frac{-1+3+22-12}{4}$

$=\frac{12}{4}=3$

$\therefore$ Remainder = 3

Thus, the remainder when p(x) is divided by g(x) is 3.

View full question & answer→

Question 23 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = 2x³ + x² - 15x - 12, g(x) = x + 2.

Answer

p(x) = 2x³ + x² - 15x - 12
g(x) = x + 2
by remainder theorem, when p(x) is divided by (x + 2), then the remainder = p(-2).
Putting x = -2 in p(x), we get
p(-2) = (-2)³ + (-2)² - 15 × (-2) - 12
= -16 + 4 + 30 - 12 = 6
$\therefore$ Remainder = 6
Thus, the remainder when p(x) is divided by g(x) is 6.

View full question & answer→

Question 33 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = x³ - 6x² + 9x + 3, g(x) = x - 1.

Answer

p(x) = x³ - 6x² + 9x + 3
g(x) = x - 1
by remainder theorem, when p(x) is divided by (x - 1), then the remainder = p(1).
Putting x = 1 in p(x), we get
p(1) = 1³ - 6 × 1² + 9 × 1 + 3
= 1 - 6 + 9 + 3 = 7
$\therefore$ Remainder = 7
Thus, the remainder when p(x) is divided by g(x) is 7.

View full question & answer→

Question 43 Marks

What must be added to 2x⁴ - 5x³ + 2x² - x - 3 so that the result is exactly divisible by (x - 2)?

Answer

Let the required number to be added be k.
Then, p(x) = 2x⁴ - 5x³ + 2x² - x - 3 + k
g(x) = x - 2
Thus, we have,
p(2) = 0
⇒ 2(2)⁴ - 5(2)³ + 2(2)² - 2 - 3 + k = 0
⇒ 32 - 40 + 8 - 5 + k = 0
⇒ k - 5 = 0
⇒ k = 5
Hence, the required number to be added is 5.

View full question & answer→

Question 53 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = x³ - 2x² - 8x - 1, g(x) = x + 1.

Answer

p(x) = x³ - 2x² - 8x - 1
g(x) = x + 1
by remainder theorem, when p(x) is divided by (x + 1), then the remainder = p(-1).
Putting x = -1 in p(x), we get
p(-1) = (-1)³ - 2 × (-1)² - 8 × (-1) - 1
= -1 - 2 + 8 - 1 = 4
$\therefore$ Remainder = 4
Thus, the remainder when p(x) is divided by g(x) is 4.

View full question & answer→

Question 63 Marks

Show that (p - 1) is a factor of (p¹⁰ - 1) and also of (p¹¹ - 1).

Answer

Let q(p) = (p¹⁰ - 1) and f(p) = (p¹¹ - 1)
By the factor theorem, (p - 1) will be a factor of q(p) and f(p) if q(1) and f(1) = 0.
Now, q(p) = p¹⁰ - 1
⇒ q(1) = 1¹⁰ - 1 = 1 - 1 = 0
Hence, (p - 1) is a factor of p¹⁰ - 1.
And, f(p) = p¹¹ - 1
⇒ f(1) = 1¹¹ - 1 = 1 - 1 = 0
Hence, (p - 1) is also a factor of p¹¹ - 1.

View full question & answer→

Question 73 Marks

The polynomials (2x³ + x² - ax + 2) and (2x³ - 3x² - 3x + a) when divided by (x - 2) leave the same remainder. Find the value of a.

Answer

Let f(x) = 2x³ + x² - ax + 2
g(x) = 2x³ - 3x² - 3x + a.
By remainder theorem, when f(x) is divided by (x - 2), then the remainder = f(2).
Putting x = 2 in f(x), we get
f(2) = 2 × 2³ + 2² - a × 2 + 2
= 16 + 4 - 2a + 2
= -2a + 22
By remainder theorem, when g(x) is divided by (x - 2), then the remainder = g(2).
g(2) = 2 × 2³ - 3 × 2² - 3 × 2 + a
= 16 - 12 - 6 + a
= -2 + a
It is given that,
⇒ -2a + 22 = -2 + a
⇒ -3a = -24
⇒ a = 8
Thus, the value of a is 8.

View full question & answer→

Question 83 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = 6x³ + 13x² + 3, g(x) = 3x + 2.

Answer

p(x) = 6x³ + 13x² + 3
g(x) = 3x + 2 $=3\Big(\text{x}+\frac{2}{3}\Big)=3\Big[\text{x}-\Big(-\frac{2}{3}\Big)\Big]$
by remainder theorem, when p(x) is divided by (3x + 2), then the remainder $=\text{p}\Big(-\frac{2}{3}\Big).$
Putting $\text{x}=-\frac{2}{3}$ in p(x), we get
$\text{p}\Big(-\frac{2}{3}\Big)=6\times\Big(-\frac{2}{3}\Big)^3+13\times\Big(-\frac{2}{3}\Big)^2+3$
$=\frac{-16+52+27}{9}$
$=\frac{63}{9}=7$
$\therefore$ Remainder = 7
Thus, the remainder when p(x) is divided by g(x) is 7.

View full question & answer→

Question 93 Marks

Using factor theorem, show that g(x) is a factor of p(x), when

p(x) = 3x³ + x² - 20x + 12, g(x) = 3x - 2

Answer

By the factor theorem, g(x) = 3x - 2 will be a factor of p(x) if $\text{p}\Big(\frac{2}{3}\Big)=0.$
Now, $\text{p}(\text{x}) = 3\text{x}^3 + \text{x}^2 - 20\text{x} + 12$
$\Rightarrow\text{p}\Big(\frac{2}{3}\Big)=3\Big(\frac{2}{3}\Big)^3+\Big(\frac{2}{3}\Big)^2-20\times\frac{2}{3}+12$
$=3\times\frac{8}{27}+\frac{4}{9}-\frac{40}{3}+12$
$=\frac{8}{9}+\frac{4}{9}-\frac{40}{3}+12$
$=\frac{8+4-120+108}{9}$
$=\frac{0}{9}$
$=0$
Hence, g(x) = 3x - 2 is a factor of the given polynomial p(x).

View full question & answer→

Question 103 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = 2x³ - 7x² + 9x - 13, g(x) = x - 3.

Answer

p(x) = 2x³ - 7x² + 9x - 13
g(x) = x - 3
by remainder theorem, when p(x) is divided by (x - 3), then the remainder = p(3).
Putting x = 3 in p(x), we get
p(3) = 2 × 3³ - 7 × 3² + 9 × 3 - 13
= 54 - 63 + 27 - 13 = 5
$\therefore$ Remainder = 5
Thus, the remainder when p(x) is divided by g(x) is 5.

View full question & answer→

Question 113 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = x³ - ax² + 6x - a, g(x) x - a.

Answer

p(x) = x³ - ax² + 6x - a,
g(x) x - a
By remainder theorem, when p(x) is divided by (x - a), then the remainder = p(a).
Putting x = a in p(x), we get
p(a) = a³ - a × a² + 6 × a - a
= a³ - a³ + 6a - a = 5a
$\therefore$ Remainder = 5a
Thus, the remainder when p(x) is divided by g(x) is 5a.

View full question & answer→

Question 123 Marks

Find the value of m for which (2x - 1) is a factor of (8x⁴ + 4x³ - 16x² + 10x + m).

Answer

Let p(x) = 8x⁴ + 4x³ - 16x² + 10x + m
It is given that (2x - 1) is a factor of p(x).
$\Rightarrow\text{p}\Big(\frac{1}{2}\Big)=0$
$\Rightarrow8\Big(\frac{1}{2}\Big)^4+4\Big(\frac{1}{2}\Big)^3-16\Big(\frac{1}{2}\Big)^2+10\times\frac{1}{2}+\text{m}=0$
$\Rightarrow8\times\frac{1}{16}+4\times\frac{1}{8}-16\times\frac{1}{4}+5+\text{m}=0$
$\Rightarrow\frac{1}{2}+\frac{1}{2}-4+5+\text{m}=0$
$\Rightarrow1+1+\text{m}=0$
$\Rightarrow2+\text{m}=0$
$\Rightarrow\text{m}=-2$

View full question & answer→

Question 133 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = x³ - 6x² + 2x - 4, $\text{g}(\text{x})=1-\frac{3}{2}\text{x}.$

Answer

p(x) = x³ - 6x² + 2x - 4
$\text{g}(\text{x})=1-\frac{3}{2}\text{x}=-\frac{3}{2}\Big(\text{x}-\frac{2}{3}\Big)$
By remainder theorem, when p(x) is divided by $\Big(1-\frac{3}{2}\text{x}\Big),$ then the remainder $=\text{p}\Big(\frac{2}{3}\Big).$
Putting $\text{x}=\frac{2}{3}$ in p(x), we get
$\text{p}\Big(\frac{2}{3}\Big)=\Big(\frac{2}{3}\Big)^3-6\times\Big(\frac{2}{3}\Big)^2+2\times\Big(\frac{2}{3}\Big)-4$
$=\frac{8}{27}-\frac{8}{3}+\frac{4}{3}-4$
$=\frac{8-72+36-108}{27}=-\frac{136}{27}$
$\therefore$ Remainder $=-\frac{136}{27}$
Thus, the remainder when p(x) is divided by g(x) is $=-\frac{136}{27}.$

View full question & answer→

Question 143 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = 3x⁴ - 6x² - 8x - 2, g(x) = x - 2.

Answer

p(x) = 3x⁴ - 6x² - 8x - 2
g(x) = x - 2
by remainder theorem, when p(x) is divided by (x - 2), then the remainder = p(2).
Putting x = 2 in p(x), we get
p(2) = 3× 2⁴ - 6 × 2² - 8 × 2 - 2
= 48 - 24 - 16 - 2 = 6
$\therefore$ Remainder = 6
Thus, the remainder when p(x) is divided by g(x) is 6.

View full question & answer→

Question 153 Marks

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where,
p(x) = 2x³ - 9x² + x + 15, g(x) = 2x - 3.

Answer

p(x) = 2x³ - 9x² + x + 15
g(x) = 2x - 3
by remainder theorem, when p(x) is divided by (2x - 3), then the remainder $=\text{p}\Big(\frac{3}{2}\Big).$
Putting $\text{x}=\frac{3}{2}$ in p(x), we get
$\text{p}\Big(\frac{3}{2}\Big)=2\times\Big(\frac{3}{2}\Big)^2+\frac{3}{2}+15$
$=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+15$
$=\frac{27-81+6+60}{4}$
$=\frac{12}{4}=3$
$\therefore$ Remainder = 3
Thus, the remainder when p(x) is divided by g(x) is 3.

View full question & answer→

Maths 3 Marks Question

Test yourself on this topic