Question

The value of k, if (x +1) is a factor of 5x3 + 3x2 – 4kx+2, is

Answer 1

Answer:

k = 0

Note:

★ If x-a is a factor of the polynomial p(x) , then x = a is its zero and hence f(a) = 0 .

Solution:

Let the given polynomial be p(x) .

Thus,

p(x) = 5x³ + 3x² - 4kx + 2

Also,

It is given that , x + 1 is a factor of the given polynomial .

Now,

If x + 1 = 0 , then

x = -1 .

Thus,

=> p(-1) = 0

{ °.° x + 1 is factor => x = -1 is a zero }

=> 5(-1)³ + 3(-1)² - 4k(-1) + 2 = 0

=> -5 + 3 + 4k + 2 = 0

=> 4k = 0

=> k = 0

Hence,

The required value of k is 0 .

Answer 2

$\Huge{\underline{\underline{ANSWER}}}$

Given :

${Cubic \: expression = 5{x}^{3}+3{x}^{2}-4kx+2}$

$\longrightarrow{Factor \: of \: the \: expression \: is \: (x + 1) }$

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Required to find :

1. Value of " k " ?

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Condition used :

➥ Factor theorem

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Explanation :

Before solving this question we need to know some content about it .

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What is a polynomial ?

A polynomial is an expression which consists of variables , coefficient generally seperated by mathematical operations such as addition or subtraction or multiplication or with non negative integral powers .

${Example : {x}^{2} - x + 6}$

Depending upon of the powers they are further divided ;

Mostly we talk about only 2

1. Quadratic expression
2. Cubic expression

Here is the example for each

Example of a quadratic expression

$\longrightarrow{{x}^{2}+7x+12} \: \: is \: an \: quadratic \: expression$

Similarly,

Example of a Cubic expression

$\longrightarrow{{z}^{3}-4z+4} \: \: \: is \: an \: cubic \: expression$

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What is a factor ?

A factor is considered to be as the multiple of the polynomial expression .

when this factor is substituted in the polynomial it gives us the remainder as zero .

When this factor value is substituted in the expression we have to do some operations by transferring the terms once it is gone .

The result is the value of K .

knowing this content is essential to solve these questions .

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Solution :

Consider the given cubic expression !

$\longrightarrow{5{x}^{3}+3{x}^{2}-4kx+2}$

Factor of the cubic expression is (x + 1)

So, let consider

${x + 1 = 0 } \\ \boxed{\implies{ x \: = - 1}}$

Now,

$p(x) = 5{x}^{3} + 3{x}^{2} - 4kx + 2$

Now, substitute the value of X here

So,

$p(-1) = 5{(-1)}^{3} + 3{(-1)}^{2} - 4k(-1) + 2$

$5(-1) + 3(1) - 4k(-1) + 2 = 0$

$-5 + 3 + 4k + 2 = 0$

$- 5 + 5 + 4k = 0$

$\cancel{ - 5 }\cancel{ + 5} + 4k = 0$

-5 and 5 gets cancelled on both sides

$4k = 0 \\ k = \dfrac{0}{4}$

As we know that any number which divides zero is also zero .

$k \: = 0$

Hence,

$\boxed{\huge{\therefore{Value \: of \: k \: = 0}}}$

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✅ Hence solved ..