Value of an Expression Using Zeros of a Quadratic Polynomial

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial

\[ p(x) = 4x^2 – 5x – 1, \]

find the value of

\[ \alpha^2\beta + \alpha\beta^2. \]

Solution

Step 1: Use Relations Between Zeros and Coefficients

For a quadratic polynomial \( ax^2 + bx + c \):

\[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \]

Here,

\[ a = 4,\quad b = -5,\quad c = -1 \]

\[ \alpha + \beta = -\frac{-5}{4} = \frac{5}{4} \]

\[ \alpha\beta = \frac{-1}{4} \]

Step 2: Simplify the Given Expression

\[ \alpha^2\beta + \alpha\beta^2 \]

Take \( \alpha\beta \) common:

\[ = \alpha\beta(\alpha + \beta) \]

Step 3: Substitute the Values

\[ \alpha^2\beta + \alpha\beta^2 = \left(-\frac{1}{4}\right)\left(\frac{5}{4}\right) \]

\[ = -\frac{5}{16} \]

Conclusion

The required value is:

\[ \boxed{-\frac{5}{16}} \]

\[ \therefore \quad \alpha^2\beta + \alpha\beta^2 = -\frac{5}{16}. \]

Value of an Expression Using Zeros of a Quadratic Polynomial

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial

\[ p(x) = 4x^2 – 5x – 1, \]

find the value of

\[ \alpha^2\beta + \alpha\beta^2. \]

Solution

Step 1: Use Relations Between Zeros and Coefficients

For a quadratic polynomial \( ax^2 + bx + c \):

\[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \]

Here,

\[ a = 4,\quad b = -5,\quad c = -1 \]

\[ \alpha + \beta = -\frac{-5}{4} = \frac{5}{4} \]

\[ \alpha\beta = \frac{-1}{4} \]

Step 2: Simplify the Given Expression

\[ \alpha^2\beta + \alpha\beta^2 \]

Take \( \alpha\beta \) common:

\[ = \alpha\beta(\alpha + \beta) \]

Step 3: Substitute the Values

\[ \alpha^2\beta + \alpha\beta^2 = \left(-\frac{1}{4}\right)\left(\frac{5}{4}\right) \]

\[ = -\frac{5}{16} \]

Conclusion

The required value is:

\[ \boxed{-\frac{5}{16}} \]

\[ \therefore \quad \alpha^2\beta + \alpha\beta^2 = -\frac{5}{16}. \]