R.S. Aggarwal solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० आयसीएसई chapter 8 - Remainder Theorem and Factor Theorem [Latest edition]

Using Remainder Theorem, find the remainder when $$f(x) = 3x^2 - 5x + 7$$ is divided by $$(x - 2)$$.

Using Remainder Theorem, find the remainder when $$f(x) = 2x^3 - 5x^2 + 3x - 10$$ is divided by $$(x - 3)$$.

Using Remainder Theorem, find the remainder when $$f(x) = 5x^3 - 12x^2 + 17x - 6$$ is divided by $$(x - 1)$$.

Using Remainder Theorem, find the remainder when $$f(x) = x^3 - 2x^2 - 5x + 6$$ is divided by $$(x + 2)$$.

Using Remainder Theorem, find the remainder when $$f(x) = 8x^3 - 16x^2 + 14x - 5$$ is divided by $$(2x - 1)$$.

Using Remainder Theorem, find the remainder when $$f(x) = 9x^2 - 6x + 2$$ is divided by $$(3x - 2)$$.

Using Remainder Theorem, find the remainder when $$f(x) = 8x^2 - 2x - 15$$ is divided by $$(2x + 3)$$.

On dividing $$(ax^3 + 9x^2 + 4x - 10)$$ by $$(x + 3)$$, we get 5 as remainder. Find the value of $$a$$.

Using Remainder Theorem, find the value of $$k$$ if on dividing $$2x^3 + 3x^2 - kx + 5$$ by $$(x - 2)$$, leaves a remainder 7.

If the polynomials $$(2x^3 + ax^2 + 3x - 5)$$ and $$(x^3 + x^2 - 2x + a)$$ leave the same remainder when divided by $$(x - 2)$$, find the value of $$a$$. Also, find the remainder in each case.

The polynomials $$f(x) = (ax^3 + 3x^2 - 3)$$ and $$g(x) = (2x^3 - 5x + a)$$ when divided by $$(x - 4)$$ leave the same remainder in each case. Find the value of $$a$$.

Find $$a$$ if the two polynomials $$ax^3 + 3x^2 - 9$$ and $$2x^3 + 4x + a$$ leave the same remainder when divided by $$(x + 3)$$.

If $$(2x^3 + ax^2 + bx - 2)$$ when divided by $$(2x - 3)$$ and $$(x + 3)$$ leaves remainders 7 and –20 respectively, find the values of $$a$$ and $$b$$.

Using the Remainder Theorem, find the remainders obtained when $$x^3 + (kx + 8)x + k$$ is divided by $$x + 1$$ and $$x - 2$$. Hence find $$k$$ if the sum of the two remainders is 1.

Using factor theorem, show that (x – 3) is a factor of \[(x^{3} + x^{2} - 17x + 15)\].

Using factor theorem, show that (x + 1) is a factor of \[(x^{3} + 4x^{2} + 5x + 2)\].

Using factor theorem, show that (3x – 2) is a factor of \[(3x^{3} + x^{2} - 20x + 12)\].

Using factor theorem, show that (3 – 2x) is a factor of \[(2x^{3} - 9x^{2} + x + 12)\].

Use factor theorem to show that (x + 2) and (2x – 3) are factors of \[(2x^{2} + x - 6)\].

Find the value of $$a$$ so that $$(x + 6)$$ is a factor of the polynomial $$(x^{3} + 5x^{2} - 4x + a)$$.

For what value of $$a$$ is the polynomial $$(2x^{3} + ax^{2} + 11x + a + 3)$$ exactly divisible by $$(2x - 1)$$?

What must be subtracted from the polynomial $$x^{3} + x^{2} - 2x + 1$$, so that the result is exactly divisible by $$(x - 3)$$?

What must be subtracted from $$16x^{3} - 8x^{2} + 4x + 7$$ so that the resulting expression has $$(2x + 1)$$ as a factor?

Using factor theorem, show that (x – 3) is a factor of \[(x^{3} - 7x^{2} + 15x - 9)\]. Hence, factorise the given expression completely.

Using factor theorem, show that (x – 4) is a factor of \[(2x^{3} + x^{2} - 26x - 40)\] and hence factorise \[(2x^{3} + x^{2} - 26x - 40)\].

Show that $$(x - 3)$$ is a factor of $$(2x^{3} - 3x^{2} - 11x + 6)$$ and hence factorise $$(2x^{3} - 3x^{2} - 11x + 6)$$.

Show that $$(3x + 2)$$ is a factor of $$(6x^{3} + 13x^{2} - 4)$$ and hence factorise $$(6x^{3} + 13x^{2} - 4)$$.

If \[2x^{3} - 3x^{2} - 3x + 2 = (2x - 1)(x^{2} + ax + b):\] 

1. using Remainder and Factor theorem, find the value of a and b.
2. hence, factorise the polynomial \[2x^{3} - 3x^{2} - 3x + 2\] completely.

While factorizing a given polynomial, using remainder and factor theorem, a student finds that $$(2x + 1)$$ is a factor of $$2x^{3} + 7x^{2} + 2x - 3$$. 

1. Is the student’s solution correct stating that $$(2x + 1)$$ is a factor of the given polynomial? 
2. Give a valid reason for your answer. Also, factorize the given polynomial completely.

Using the factor theorem, show that (x – 2) is a factor of \[x^{3} + x^{2} - 4x - 4\]. Hence factorise the polynomial completely.

If $$(x - 2)$$ is a factor of $$2x^3 - x^2 - px - 2$$, 

1. find the value of $$p$$ 
2. with the value of $$p$$, factorise the above expression completely.

Find the value of $$a$$, if $$(x - a)$$ is a factor of the polynomial $$3x^3 + x^2 - ax - 81$$.

Find the values of a and b, if (x – 1) and (x + 2) are both factors of \[(x^{3} + ax^{2} + bx - 6)\].

If (x + 2) and (x + 3) are factors of \[x^{3} + ax + b\], find the values of a and b.

If $$(x^3 + ax^2 + bx + 6)$$ has $$(x - 2)$$ as a factor and leaves a remainder 3 when divided by $$(x - 3)$$, find the values of $$a$$ and $$b$$.

Using factor theorem, factorise the following:

\[x^{3} + 7x^{2} + 7x - 15\]

Using factor theorem, factorise the following:

\[6x^{3} - 7x^{2} - 11x + 12\]

Using factor theorem, factorise the following:

\[2x^{3} + 3x^{2} - 9x - 10\]

Using factor theorem, factorise the following:

\[2x^{3} + 19x^{2} + 38x + 21\]

Using factor theorem, factorise the following:

\[3x^{3} + 2x^{2} - 19x + 6\]

Using factor theorem, factorise the following:

\[2x^{3} + x^{2} - 13x + 6\]

Using factor theorem, factorise the following:

\[2x^{3} - x^{2} - 13x - 6\]

If (x – 2) is a factor of \[(x^{3} + 2x^{2} - kx + 10)\], find the value of k. Hence, determine whether (x + 5) is also a factor of the given expression. 

[Hint : Let \(f(x) = x^{3} + 2x^{2} - kx + 10\). Then, f(2) = 0, gives k = 13.
Thus, \(f(x) = x^{3} + 2x^{2} - 13x + 10\). Now, show that f(–5) = 0.]

Using the Remainder and Factor Theorems, factorise the polynomial. \(x^3 + 10x^2 - 37x + 26.\) 

[Hint : Let \(p(x) = x^3 + 10x^2 - 37x + 26\). Then, \(p(1) = 0\).
\(\therefore (x - 1)\) is a factor of \(p(x)\).]

If (x – 2) is a factor of the expression \(2x^3 + ax^2 + bx - 14\) and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b. 

[Hint : Let \(p(x) = 2x^3 + ax^2 + bx - 14\). Then, \(p(2) = 0\) and \(p(3) = 52\).]

The polynomial 3x³ + 8x² – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

It is given that (x − 2) is a factor of polynomial 2x³ − 7x² + kx − 2.

Find:

1. the value of ‘k’. 
2. Hence, factorise the resulting polynomial completely.

A polynomial in ‘x’ is divided by (x – a) and for (x – a) to be a factor of this polynomial, the remainder should be ______.

If $$p(x) = x + 4$$, then $$p(x) + p(-x) = ?$$

If $$p(x) = x^2 - 2\sqrt{2}x + 1$$, $$p(2\sqrt{2}) = ?$$

If $$f(x) = 3x - 5x^2 - 1$$, then $$f(-1) = ?$$

If $$(x^{101} + 101)$$ is divided by $$(x + 1)$$, then the remainder is ______.

If $$(3x^3 - 5x^2 + 3x - 7)$$ is divided by $$(x - 2)$$, then the remainder is ______.

If $$f(x) = x^4 - ax^3 + x^2 - ax$$ is divided by $$(x - a)$$, then the remainder is ______.

When $$f(x) = x^3 + ax^2 + 2x + a$$ is divided by $$(x + a)$$, then the remainder is ______.

If $$(x^2 - 7x + a)$$ leaves a remainder 1 when divided by $$(x + 1)$$, then the value of $$a$$ is ______.

In the division of a cubic polynomial $$f(x)$$ by a linear polynomial, the remainder is $$f(-2)$$, then the divisor must be ______.

If $$p(t) = t^2 - t - 2$$, then the value of $$p\left(-\frac{1}{3}\right)$$ is ______.

If $$f(x) = 4x^3 - 3x^2 + 5$$ is divided by $$(2x + 1)$$, then the remainder is ______.

If $$(x^{50} - 1)$$ is divided by $$(x - 1)$$, then the remainder is ______.

When $$2x^2 - 3kx + k$$ is divided by $$(x + 2)$$, then the remainder obtained is 5. The value of $$k$$ is ______.

If the polynomial $$p(x) = x^3 - 4kx + 3$$ is divided by $$(2x + 1)$$, then the remainder obtained is –3. The value of $$k$$ is ______.

If $$(x + 5)$$ is a factor of $$f(x) = x^3 - 20x + 5k$$, then $$k = ?$$

For what value of $$k$$ is the polynomial $$f(x) = 2x^3 - kx^2 + 3x + 10$$ exactly divisible by $$(x + 2)$$?

If $$(x^{50} + 2x^{49} + k)$$ is divisible by $$(x + 1)$$, then the value of $$k$$ is ______.

The value of $$p$$ for which $$(x - p)$$ is a factor of $$x^3 - px^2 + x + 5$$ is ______.

If $$(x - 2)$$ is a factor of $$x^3 - kx - 12$$, then the value of $$k$$ is ______.

For a polynomial $$f(x)$$, $$f(-1)$$ and $$f(2)$$ are both equal to zero. Which of the following is a factor of $$f(x)$$?

$$(x + 1)$$ is a factor of the polynomial :

A polynomial in $$x$$ is $$x^3 + 5x^2 - kx - 24$$. Which of the following is a factor of the given polynomial so that the value of $$k$$ is 2?

What number should be subtracted from $$2x^3 - 5x^2 + 5x$$, so that the resulting polynomial has $$(2x - 3)$$ as a factor?

If $$(x^2 + ax + b)$$ is divided by $$(x + c)$$, then the remainder is ______.

$$(x - 2)$$ and $$(x + 2)$$ are the factors of $$x^3 + x^2 - 4x - 4$$. The third factor of the given polynomial is ______.

For two polynomials $$f(x)$$ and $$g(x)$$, $$(x - a)$$ and $$(x - b)$$ are their respective factors. Which of the following is true?

If the polynomial 2x³ + 3x² – 2x – 3 is completely divisible by (2x + a) and the quotient is equal to (x² – 1), then one of the values of a is ______.

Assertion (A): When a polynomial $$f(x)$$ is divided by $$(3x + 4)$$, then the remainder is $$f\left(-\frac{3}{4}\right)$$.

Reason (R): Remainder theorem states that when a polynomial $$f(x)$$ is divided by $$(x - \alpha)$$, then the remainder is $$f(\alpha)$$.

Assertion (A): $$(x - 1)$$ is a factor of $$x^3 + 2x^2 - x - 2$$.

Reason (R): If $$(x + \alpha)$$ is a factor of $$f(x)$$, then $$f(\alpha) = 0$$.

Assertion (A): If $$(2x - 1)$$ is a factor of polynomial $$f(x)$$, then $$f\left(\frac{1}{2}\right) = 0$$. 

Reason (R): $$(ax + b)$$ is a factor of $$f(x)$$ implies $$f\left(-\frac{b}{a}\right) = 0$$.

Assertion (A): $$x^3 + 2x^2 - x - 2$$ is a polynomial of degree 3.

Reason (R): $$x + 2$$ is a factor of the polynomial.

Pamela factorized the following polynomial:

2x³ + 3x² – 3x – 2

She found the result as (x + 2) (x – 1) (x – 2). Using remainder and factor theorem, verify whether her result is correct. If incorrect, give the correct result.

Using remainder and factor theorem, show that (2x + 3) is a factor of the polynomial 2x² + 11x + 12. Hence, factorise it completely. What must be multiplied to the given polynomial so that x² + 3x – 4 is a factor of the resulting polynomial? Also, write the resulting polynomial.

Given, $$9x^2 - 4$$ is a factor of $$9x^3 - mx^2 - nx + 8$$. 

1. find the values of $$m$$ and $$n$$ using the remainder and factor theorem 
2. factorise the given polynomial completely.

If a polynomial x³ + 2x² – ax + b leaves a remainder –6 when divided by x + 1 and the same polynomial has x – 2 as a factor, then find the values of a and b.