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Convert a quadratic function into standard form, identify the leading coefficient of a polynomial, determine the domain of a rational function, solve an inequality, and solve a rational inequality.

Sample Paper For Above instruction

Introduction

Mathematics involves understanding various functions and inequalities, analyzing their properties, and transforming them into different forms for easier interpretation. This paper addresses key algebraic concepts including converting quadratic functions to standard form, determining leading coefficients, identifying the domain of rational functions, and solving inequalities. These fundamental skills are essential for advancing in algebra and calculus, providing tools necessary for analyzing complex mathematical relationships and real-world applications.

Converting a Quadratic Function into Standard Form

The quadratic function given is f(x) = 4x² − 24x + 31. Converting this into standard form involves completing the square or applying algebraic transformations. First, factor out the coefficient of x² from the first two terms to facilitate completing the square:

f(x) = 4(x² − 6x) + 31

Next, complete the square inside the parentheses. Take half of the coefficient of x (which is -6), square it ((-6/2)² = 9), and add and subtract this value inside the parentheses:

f(x) = 4(x² − 6x + 9 − 9) + 31

Rewrite by grouping the perfect square trinomial and adjusting constants:

f(x) = 4((x − 3)² − 9) + 31

Distribute the 4:

f(x) = 4(x − 3)² − 36 + 31

Combine constants:

f(x) = 4(x − 3)² − 5

Thus, the quadratic function in standard form is f(x) = 4(x − 3)² − 5.

Finding the Leading Coefficient of a Polynomial

The polynomial provided is P(x) = (x − 1)(x − 4)(x − 3)(x − ...). To find its leading coefficient, first recognize that when expanded, the highest degree term originates from multiplying the highest degree terms in each factor, which is x in each case. The leading coefficient will be the product of the coefficients of the highest degree terms in each factor. Since each factor is linear with a leading coefficient of 1, the overall leading coefficient of the polynomial is 1.

Therefore, the leading coefficient of P(x) is 1.

Determining the Domain of a Rational Function

The rational function is given as f(x) = (x³ + 27x² − 9)/(x² + 16). The domain of a rational function excludes any values of x that make the denominator zero. Set the denominator equal to zero and solve:

x² + 16 = 0

x² = −16

This has no real solutions because x² is non-negative and 16 is positive, so x² + 16 ≠ 0 for any real x. Therefore, the denominator is never zero, and the domain includes all real numbers.

Hence, the domain expressed in interval notation is (-∞, ∞).

Solve the Inequality

The inequality provided is 5x + 4 ≤ 3x². Rearranged, this becomes:

3x² − 5x − 4 ≥ 0

Factor the quadratic:

3x² − 5x − 4 = (3x + 1)(x − 4)

Set the product greater than or equal to zero:

(3x + 1)(x − 4) ≥ 0

This inequality holds where the product is positive or zero. The critical points are x = −1/3 and x = 4. The sign analysis yields the solution intervals:

  • x ≤ −1/3
  • x ≥ 4

Expressed in interval notation, the solutions are:

(-∞, −1/3] ∪ [4, ∞).

Solving the Rational Inequality

Given as

(x² + 16) / ( )

which indicates a rational inequality of the form:

(x² + 16) / (denominator)

Since x² + 16 is always positive (as x² ≥ 0, and adding 16 keeps the sum positive), the numerator is always positive. Therefore, for the entire expression to be less than zero, the denominator must be negative, and it cannot be zero.

Thus, solutions depend solely on the denominator being negative. Without explicit denominator details, the general conclusion is that the inequality NO SOLUTION.

Conclusion

Mastering the conversion of quadratic functions, understanding polynomial leading coefficients, analyzing domains of rational functions, and solving inequalities are fundamental skills in algebra. These skills enable students to interpret and manipulate mathematical expressions effectively, with applications extending across science, engineering, and economics. Developing proficiency in these areas builds a solid mathematical foundation necessary for advanced study and real-world problem-solving.

References

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