Question

find the zeroes of the given polynomial 1) p(X) = x⁴-16​

Answer 1

Answer:

Zero of the given polynomial are - +2, +2, -2, -2

Step-by-step explanation:

Given:-

$\sf \implies p(x) = x^4-16$

To Find:-

Zero of the polynomial.

Procedure:-

$\sf \implies x^2 - 16 = 0$

Add 6 to both sides:-

$\implies \sf x^2 -16 +16 = 0 +16$

Simplifying further:-

$\sf \implies x^4 = 16$

Taking root:-

$\sf \implies x = \pm(16)^{\dfrac{1}{4}}$

$\sf \implies x = +2, -2$

Since, the polynomial is Bi-Quadratic, it's must to have 4 zeroes. We know two of it (+2) and (-2).

so, the zeroes should be. (+2) , (+2) , (-2) and (-2).

Verification:-

let's substitute x = -2 in the given polynomial.

$\implies \sf p(-2) = (-2)^4 -16$

$\implies \sf p(-2) = 16 - 16$

$\implies \sf p(-2) = 0$

hence, proved that (-2) is the zero of polynomial.

Now, let's substitute p(2) in the given polynomial.

$\implies \sf p(2) = (2)^4 -16$

$\implies \sf p(2) = 16 -16$

$\implies \sf p(2) = 0$

hence, proved that (2) is the zero of polynomial.

NotE:-

• Polynomial : A variable bases language in maths. or A string of variables and numbers put together.

• Degree of polynomial : Highest power of a polynomial is called degree of polynomial.

• Coefficient : A number by which a variable is multiplied.

• Constant : It's a numerical term holding no variable.

Answer 2

Answer :-

☞ Zeroes of x⁴ - 16 = +2, +2, -2 , -2

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★ Concept :-

Here the concept of Bi - Quadratic Polynomial is used. Bi - Quadratic polynomial is a type of polynomial which has 4 roots or 4 zeroes as its solution and the highest degree of this type of polynomial is 4. Even, when drawing the graph of this polynomial, intersects x - axis at 4 places.

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★ Solution :-

Given,

▶ p(x) = x⁴ - 16

In order to find zeroes of this polynomial,we must use Factor Theorem, right ?

Yes, according to factor theorem, this equation must be equal to 0 . So let us apply it.

✒ x⁴ - 16 = 0

✒ x⁴ = 16

✒ x = ± ⁴√16 = ± √4

✒ x = +2 , - 2

Since, this is a Bi Quadratic Polynomial, as discussed above, it must have 4 zeroes. But here we are getting finally two zeroes.. So our result should be four zeroes that is repeating two zeroes because other than them nothing is zero of the given equation.

✳ Zeroes of (x⁴ - 16) = +2 , +2 , -2 , -2

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★ Verification :-

In order to verify this given p(x) we can simply apply the values we got.

~ Case I :-

✏ p(x) = x⁴ - 16

✏ p(2) = (2)⁴ - 16

✏ 16 - 16 = 0

✏ 0 = 0

Clearly, LHS = RHS .

~ Case II :-

✏ p(x) = x⁴ - 16

✏ p(-2) = (-2)⁴ - 16

✏ 16 - 16 = 0

✏ 0 = 0

Clearly, LHS = RHS.

Here both the conditions satisfy, so our answer is correct.

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★ More to know :-

• Linear Polynomial is the type of polynomial which has highest degree of 1 and intersects x - axis once when drawn as graph and gives single zero.

Standard Form :- ax + b = 0

• Quadratic Polynomial is the type of polynomial which has highest degree of 2 and intersects x-axis twice when drawn as graph and gives two zeroes.

Standard Form :- ax² + bx + c = 0

• Cubic Polynomial is the type of polynomial which has highest degree of 3 and intersects x-axis thrice when drawn as graph and gives three zeroes.

Standard Form :- ax³ + bx² + cx + d = 0

• Polynomial is a equation formed by using constants and Variables. Constants are the numeric values and variables are alphabetical values for which we solve the equation.