Finding the Value of k in a Quadratic Polynomial

We are given a quadratic polynomial \(f(x) = x^2 - 8x + k\). We are also told that the sum of the squares of the zeros of this polynomial is 40. We need to find the value of the constant term, \(k\).

Let the zeros of the quadratic polynomial \(f(x)\) be \(\alpha\) and \(\beta\). For a standard quadratic polynomial \(ax^2 + bx + c\), the relationships between the zeros and the coefficients are:

• Sum of zeros: \(\alpha + \beta = -\frac{b}{a}\)
• Product of zeros: \(\alpha \beta = \frac{c}{a}\)

In our given polynomial \(f(x) = x^2 - 8x + k\), the coefficients are \(a=1\), \(b=-8\), and \(c=k\). Using the relationships above, we can find the sum and product of the zeros:

• Sum of zeros: \(\alpha + \beta = -\frac{(-8)}{1} = 8\)
• Product of zeros: \(\alpha \beta = \frac{k}{1} = k\)

We are given that the sum of the squares of the zeros is 40. This means:

\(\alpha^2 + \beta^2 = 40\)

We know an algebraic identity that relates the sum of squares of two numbers to their sum and product: \((\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta\). We can rearrange this identity to express the sum of squares in terms of the sum and product:

\(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)

Now, we can substitute the values we found for \((\alpha + \beta)\) and \((\alpha \beta)\) into this equation, along with the given value for \((\alpha^2 + \beta^2)\):

\(40 = (8)^2 - 2(k)\)

Let's simplify and solve for \(k\):

\(40 = 64 - 2k\)

Now, we isolate the term with \(k\):

\(2k = 64 - 40\)

\(2k = 24\)

Finally, divide by 2 to find the value of \(k\):

\(k = \frac{24}{2}\)

\(k = 12\)

Thus, the value of \(k\) for the quadratic polynomial \(f(x) = x^2 - 8x + k\) such that the sum of the squares of its zeros is 40, is 12.

Revision Table: Key Concepts for Quadratic Polynomials

Additional Information: Relating Zeros and Coefficients

The relationship between the zeros and coefficients of a polynomial is a fundamental concept in algebra. For a quadratic polynomial \(ax^2 + bx + c = 0\), these relationships are derived from Vieta's formulas. These formulas are incredibly useful for solving problems involving the zeros without actually finding the values of the zeros themselves, as demonstrated in this problem where we used the sum and product of zeros to find a missing coefficient \(k\).

The identity \((\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta\) is a basic algebraic identity that is frequently used in problems involving the sum of squares of roots. It allows us to connect the sum of squares to the sum and product of the roots, which are directly related to the coefficients of the polynomial.