P(x)=x^2 + 2(√2)x - 6 Find the Zero of the Polynomial and Verify between Zero and Coefficients

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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