Question

If \(\alpha\) and \(\beta\) are the zeros of the quadratic polynomial \(a x^{2}+b x+c\) and \(x\) lies between \(\alpha\) and \(\beta\), then which of the following is true? (1) If \(\mathrm{a}<0\) then \(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}>0 .\) (2) If \(\mathrm{a}>0\) then \(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}<0 .\) (3) If \(\mathrm{a}>0\) then \(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}>0\). (4) Both (1) and (2).

Short answer

The correct statement about the quadratic polynomial \(ax^2 + bx + c\) when \(x\) lies between its zeros \(\alpha\) and \(\beta\) is: (1) If \(a < 0\) then \(ax^2 + bx + c > 0\)

Step-by-step solution

Write down quadratic polynomials in terms of its zeros

Given \(\alpha\) and \(\beta\) as zeros of \(a x^{2}+b x+c\), we can write the quadratic polynomial as: $$a (x -\alpha)(x -\beta)$$ Now, we need to analyze the behavior of the polynomial in between the zeros.

Determine the sign of the quadratic polynomial in between the zeros

As x lies between \(\alpha\) and \(\beta\), we have: \(\alpha < x < \beta\) Now consider the two cases, \(\alpha > 0\) and \(\alpha < 0\), separately. Case 1: \(\alpha > 0\) and \(\beta > 0\) If \(a > 0\), the term \((x - \alpha)\) would be negative and \((x - \beta)\) would be negative, so the product \(a(x - \alpha)(x - \beta)\) would be positive. If \(a < 0\), the term \((x - \alpha)\) would be negative and \((x - \beta)\) would be negative, so their product multiplied by a negative a would also be positive. Case 2: \(\alpha < 0\) and \(\beta < 0\) This case can't happen since \(\alpha\) and \(\beta\) are zeros of the same quadratic polynomial, meaning one must be positive and the other negative. So, the analysis shows that if x lies between the zeros of the quadratic polynomial, the polynomial will be positive. Thus, the true statement is: (1) If \(a<0\) then \(ax^{2}+bx+c>0\) So, the answer is (1).

Key concepts

Zeros of a Quadratic Polynomial

In a quadratic polynomial, often expressed in the form \( ax^2 + bx + c \), the zeros, also called roots, are the values of \( x \) where the polynomial equals zero. These zeros are usually denoted as \( \alpha \) and \( \beta \). If you set the quadratic polynomial equal to zero, it can be factored to \( a(x-\alpha)(x-\beta) \). By solving this equation, you identify the zeros \( \alpha \) and \( \beta \).

• The zeros are crucial as they define critical points on the graph of the quadratic.
• The location of zeros helps in shaping and positioning the parabola.

Understanding zeros helps us know where the quadratic touches or crosses the x-axis and determines the vertex's position and the parabola's direction. Consider whether the zeros are real, distinct, or complex as this affects the entire quadratic behavior.

Polynomial Behavior Between Roots

The behavior of a quadratic polynomial between its zeros \( \alpha \) and \( \beta \) is integral to understanding how the polynomial behaves. When you factor a quadratic into \( a(x-\alpha)(x-\beta) \), consider the interval \( \alpha < x < \beta \).

• For \( a > 0 \), the parabola opens upwards and the quadratic typically takes negative values between \( \alpha \) and \( \beta \).
• For \( a < 0 \), the parabola opens downwards, and the quadratic takes positive values in this interval.
• In simpler terms, if \( x \) lies between the zeros and \( a > 0 \), the value of the polynomial is negative; if \( a < 0 \), it is positive.

This pattern occurs because the product of \( (x-\alpha) \) and \( (x-\beta) \) depends on how they interact with \( a \). Reflecting this behavior on a graph shows dips or rises depending on \( a \).

Sign of Quadratic Expressions

The sign of quadratic expressions, specifically \( ax^2 + bx + c \), indicates whether the expression is positive or negative over an interval. This expression's sign is determined by the leading coefficient \( a \) and is crucial between zeros \( \alpha \) and \( \beta \).

• If \( a > 0 \), then between \( \alpha \) and \( \beta \), \( ax^2 + bx + c \) is often negative, reaching its minimum at the midpoint of the interval.
• If \( a < 0 \), the situation is reversed, meaning it is positive between the zeros.
• Understanding this sign change helps predict and solve problems involving quadratic inequalities and optimizations.

Thus, assessing the sign of the quadratic form aids in predicting graph behavior, such as concavity and whether a point lies above or below the x-axis. It serves as an essential factor in determining the nature and features of quadratics.