Question

Find all the zeros of p bracket x is equal to X ki power 4 - 3 x cube minus 5 x square + 21 x minus 14 if 2 its zeros are root 7 and minus root 7

Answer 1

Question

Find all the zeros of p(x) = x⁴ - 3x³ - 5x² + 21x - 14. If two its zeros are √7 and - √7.

Solution

Given that, two of the zeros are √7 and -√7. So, factors are (x - √7) and (x + √7).

→ x² - (√7)²

→ x² - 7 or x² - 0x - 7

x² -0x-7 ) x⁴ - 3x³ - 5x² + 21x - 14 (x²-3x+2

..............-(x⁴ - 0x³ - 7x²)

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................0 - 3x³ + 2x² + 21x

................. - (-3x³ + 0x² + 21x)

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............................. 2x² - 14

.......................... - (2x² - 14)

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...................................0

Now,

Dividend = Quotient × Divisor

p(x) = (x - √7)(x + √7)(x² - 3x + 2)

→ (x - √7)(x + √7)(x² - 2x - x + 2)

→ (x - √7)(x + √7)[x(x - 2) -1(x - 2)]

→ (x - √7)(x + √7)(x - 1)(x - 2)

So, zeros of p(x) are √7, -√7, 1 and 2

Answer 2

Given Polynomial :-

• x⁴ - 3x³ - 5x² + 21x - 14 = 0

Two Zeros :-

• √7 & (-√7) .

Concept Used :-

• if a & b are the Zeros of The Polynomial , Than, (x +a) & (x + b) are The factors of Given Polynomial.

Solution :-

with Above Told Concept we can say That, (x +√7) & (x - √7) are the Factors of Given Polynomial.

or,

→ (x + √7)(x - √7) = x² - 7 { using (a + b)(a - b) = a² - b² }.

So, (x² - 7) will be a factor of Given Polynomial.

Hence, Dividing The Given Polynomial with (x² - 7) we will get Remainder Zero.

⟪ Check The Divide Part in Image Now. ⟫

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From image we can see That, we get remainder Zero,

and,

→ Quotient = x² - 3x + 2.

Putting This Equal to Zero, and Than, solving by Splitting The Middle Term, we get,

→ x² - 3x + 2 = 0

→ x² - 2x - x + 2 = 0

→ x(x - 2) - 1(x - 2) = 0

→ (x - 2) (x - 1) = 0

Putting both Equal to Zero now,

→ x - 2 = o

→ x = 2

Or,

→ x - 1 = 0

→ x = 1.

Hence, All Zeros of The Given Polynomial are { 1, 2 , √7 & (-√7) } .