Question

show that (x-2) and (x-3)
(x-3) are factors off (x) =x³–3x²–10x+24

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Answer 1

Given question -: To show that

$(x - 2)(x - 3) \: are \: factor \: of \: {x}^{3} - 3 {x}^{2} - 10x + 24$

let us first understand the meaning of factor theorem

According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and 'a' is any real number then, (x-a) is a factor of f(x), if f(a)=0.

Now

let's solve the question

according to factor theorem

(x - 2 ) = 0

x = 2

now put the value of x in given equation

${x}^{3} - 3 {x}^{2} - 10x + 24$

${2}^{3} - 3 {}^{2} - 10 \times 2 - 24$

$8 - 9 - 20 + 24$

$- 1 - 20 + 24$

$- 21 +24$

hence x = 2 is not a factor of given equation

Again

x -3 =0

x = 3

Now put x = 3 in given equation

$x ^{3} - 3 {x}^{2} - 10x + 24$

$3 {}^{3} - 3 {}^{2} - 10 \times 3 + 24$

$27 - 9 - 30 + 24$

$27 - 39 +24$

$27 - 15$

12

hence x = 3 is also not a factor of given equation

Hope this will help you