Question

If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(a x^{2}+b x+c\) such that \(x\) does not lie between \(\alpha\) and \(\beta\), then (1) \(a>0\) and \(a x^{2}+b x+c<0\) (2) \(a x^{2}+b x+c \leq 0\) and \(a<0\) (3) \(\mathrm{a}>0\) and \(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}>0\) (4) both (2) and (3).

Short answer

(1) a > 0 and ax^2 + bx + c < 0 (2) ax^2 + bx + c <= 0 and a < 0 (3) a > 0 and ax^2 + bx + c > 0 (4) Both (2) and (3) Answer: (1) a > 0 and ax^2 + bx + c < 0

Step-by-step solution

Evaluate the polynomial at the zeroes

Since \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(ax^2 + bx + c\), we have $$ a\alpha^2 + b\alpha + c = 0 $$ and $$ a\beta^2 + b\beta + c = 0 $$

Compare polynomial values at \(\alpha\) and \(\beta\)

Since \(x\) does not lie between \(\alpha\) and \(\beta\), then both \(ax^2 + bx + c\) evaluated at \(\alpha\) and \(\beta\) should have the same sign. Let's write an inequality for this. $$ a\alpha^2 + b\alpha + c >0 > a\beta^2 + b\beta + c \ \ (\text{if} \ a > 0) $$ or $$ a\alpha^2 + b\alpha + c <0 < a\beta^2 + b\beta + c \ \ (\text{if} \ a < 0) $$

Determine the correct option

Now we look at the given options and compare them to the inequalities we derived in step 2. (1) \(a > 0\) and \(ax^2 + bx + c < 0\) This option corresponds to the first inequality we derived in step 2, so it is a possible answer. (2) \(ax^2 + bx + c \leq 0\) and \(a < 0\) This option is slightly different from the second inequality we derived in step 2, because it has "\(\leq 0\)" instead of "\(< 0\)", meaning it might be true if \(x=\alpha\) or \(x=\beta\). This cannot be true since \(x\) must not lie between or coincide with \(\alpha\) or \(\beta\). Therefore, this option is incorrect. (3) \(a > 0\) and \(ax^2 + bx + c > 0\) This option is opposite to the first inequality we derived in step 2, so it is incorrect. (4) Both (2) and (3) Since option (3) is incorrect, this option is also incorrect. #Conclusion# The correct answer is (1) \(a > 0\) and \(ax^2 + bx + c < 0\).

Key concepts

Roots of Polynomials

When dealing with quadratic polynomials like \(a x^2 + bx + c\), it's crucial to understand the concept of roots, often referred to as zeroes. The roots of a quadratic polynomial are the values of \(x\) for which the polynomial equals zero. These are denoted by \(\alpha\) and \(\beta\) for a quadratic function. In simple terms, these are the points where the graph of the polynomial touches or crosses the x-axis.

If given that \(\alpha\) and \(\beta\) are roots, it implies:

• \(a\alpha^2 + b\alpha + c = 0\)
• \(a\beta^2 + b\beta + c = 0\)

These equations are used to determine the characteristics of the polynomial, such as being able to evaluate where the polynomial holds certain values. In the context of quadratic polynomials, these roots help us analyze the intervals over which the polynomial is positive or negative.

Quadratic Equations

A quadratic equation is a mathematical expression of the form \(ax^2 + bx + c = 0\). It is a type of polynomial equation that features a second-degree term. The solutions of this equation (i.e., the values of \(x\)) can be found using the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

One important aspect of solving quadratic equations involves determining whether or not real roots exist. This is done by analyzing the discriminant, \(b^2 - 4ac\):

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, and it's called a repeated root.
• If \(b^2 - 4ac < 0\), there are no real roots, only complex ones.

Understanding quadratic equations is vital for distinguishing between the roots' characteristics and interpreting their graphical representation.

Polynomial Inequalities

Examining polynomial inequalities involves analyzing the regions where a polynomial is positive, negative, or zero. For quadratic polynomials, this is primarily about determining where the expression \(ax^2 + bx + c\) satisfies a given inequality (e.g., \(>0\), \(<0\), \(\leq 0\), or \(\geq 0\)).

Graphically, this corresponds to finding out where the parabola lies above or below the x-axis. Consider the quadratic polynomial \(ax^2 + bx + c\):

• If \(a > 0\), the parabola opens upwards, and it is negative between its roots if the expression \(<0\).
• If \(a < 0\), the parabola opens downwards, and it is positive between its roots if the expression \(>0\).

By analyzing the sign of the expression depending on its coefficients and roots, we can determine the intervals that satisfy the polynomial inequality. This process helps solve problems where the solution set for \(x\) is required to lie outside certain regions defined by the roots.