P(x)=x^2 + 2(√2)x – 6 Find the Zero of the Polynomial and Verify between Zero and Coefficients
By BYJU'S Exam Prep
Updated on: October 17th, 2023
P(x)=x2 + 2(√2)x – 6 find the zero of the polynomial and verify between zero and coefficients
Here are the steps to find the zeros of the quadratic polynomial P(x) = x2 + 2(√2)x – 6
- We can find out the zeros by using quadratic formula(x = (-b ± √(b2 – 4ac)) / (2a)) or by the method shown below.
- On simplifying by either of the methods, we will get the zeros.
- To verify the relationship between the zeros and the coefficients we can use Vieta’s formulas. Let us explore more in the coming section.
Table of content
P(x)=x2 + 2(√2)x – 6 Find the Zero of the Polynomial and Verify between Zero and Coefficients
Solution:
Let’s solve the quadratic polynomial P(x) = x2 + 2(√2)x – 6 using a different method, factoring.
To factor the quadratic polynomial, we need to find two numbers that multiply to give -6 (the coefficient of the constant term) and add up to 2(√2) (the coefficient of the linear term).
The two numbers that meet these conditions are √6 and -√6. Therefore, we can rewrite the polynomial as:
P(x) = (x + √6)(x – √6)
Setting this expression equal to zero, we have:
(x + √6)(x – √6) = 0
Now we can apply the zero product property, which states that if a product of factors is equal to zero, then at least one of the factors must be zero.
Setting each factor equal to zero, we get:
x + √6 = 0 –> x = -√6
x – √6 = 0 –> x = √6
Therefore, the zeros of the quadratic polynomial P(x) = x2 + 2(√2)x – 6 are x = -√6 and x = √6.
Now, let’s verify the relationship between the zeros and the coefficients using Vieta’s formulas:
The sum of the zeros is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term:
Sum of zeros = -2(√2) / 1 = -2√2
The product of the zeros is equal to the constant term divided by the coefficient of the quadratic term:
Product of zeros = -6 / 1 = -6
Indeed, the sum of the zeros, -2√2, matches the coefficient of the linear term, and the product of the zeros, -6, matches the constant term. Therefore, the relationship between the zeros and the coefficients is verified.
Answer:
Zero of the Polynomial P(x)=x2 + 2(√2)x – 6 are x = -√6 and x = √6 and the Relation between Zero and Coefficients Verfied Correctly
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