Question

q.25
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Answer 1

Step-by-step explanation:

Given: 3x³-x²-3x+1

To find: Factors using factor theorem.

Solution:

We know that if x-a is a factor of polynomial p(x), then f(a)=0.

Step 1: Analyze the given cubic polynomial

Put x=1

$p(x) = 3 {x}^{3} - {x}^{2} - 3x + 1 \\ \\ p(1) = 3( {1)}^{3} - ( {1)}^{2} - 3(1) + 1 \\ \\ p(1) = 3 - 1 - 3 + 1 \\ \\ p(1)= 0$

Thus,

(x-1) is a factor of p(x).

Step 2: Divide p(x) by (x-1)

$x - 1 \: )3 {x}^{3} - {x}^{2} - 3x + 1(3 {x}^{2} + 2x - 1\\ 3 {x}^{3} - 3 {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \\ ( - ) \: \: \: \: ( + ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - - \\ 2 {x}^{2} - 3x \\ 2 {x}^{2} - 2x \\ ( - ) \: \: ( + ) \\ - - - - - - - - - \\ - x + 1 \\ - x + 1 \\ ( + ) \: \: ( - ) \\ - - - - - - \\ 0 \\ - - - - - -$

Step 3: Find factors of quotient polynomial

$3 {x}^{2} + 2x - 1 \\ \\ 3 {x}^{2} + 3x - x - 1 \\ \\ 3x(x + 1) - 1(x + 1) \\ \\ (3x - 1)( x + 1)$

Step 4: Factors of polynomial are

$(x - 1)(x + 1)(3x - 1)$

Final answer:

$\bold{\red{3{x}^{3} -{x}^{2}-3x - 1 =(x - 1)(x + 1)(3x - 1)}}$

Hope it helps you.

To learn more on brainly:

solve the pair of equations 0.2x + 0. 3y = 1.3 and 0.4x + 0.5y = 2.3 by substitution method .

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https://brainly.in/question/44915806