Question

The quadratic equation \(p(x)=0\) with real coefficients has purely imaginary roots. Then the equation $$ p(p(x))=0 $$ has (A) only purely imaginary roots (B) all real roots (C) two real and two purely imaginary roots (D) neither real nor purely imaginary roots

Short answer

The equation has all real roots (B).

Step-by-step solution

Understanding the Given Quadratic Equation

If a quadratic equation with real coefficients has purely imaginary roots, then it can be written in the form of \( ax^2 + bx + c = 0 \) where \( b^2 < 4ac \) which ensures that the discriminant is negative. Hence, the roots are of the form \( bi \) and \( -bi \), where \( b eq 0 \) and \( i \) is the imaginary unit.

Analyzing the Nature of Roots

For the quadratic equation \( p(x) = 0 \) with roots \( bi \) and \( -bi \), we can express \( p(x) = a(x - bi)(x + bi) \). Expanding this, we get \( p(x) = ax^2 + a(b^2) \). Since we're considering the composition of the function with itself, \( p(p(x)) = 0 \), we have to substitute \( x \) in \( p(x) \) with \( p(x) \) itself.

Composing Function with Itself

Composing \( p(x) \) with itself gives \( p(p(x)) = a(ax^2 + a(b^2))^2 + a(b^2) \). This simplifies to an equation of the form \( a^3x^4 + Ca = 0 \), where \( C \) is some constant derived from expanding the squared term and \( a eq 0 \).

Determining the Roots of the Composed Equation

The equation \( a^3x^4 + Ca = 0 \) is a quartic equation. Since all coefficients are real, the non-real roots must come in conjugate pairs. Given that the original quadratic has purely imaginary roots, and we are squaring these roots in the equation \( p(p(x)) \), the resulting roots must be non-negative real numbers, specifically squares of the imaginary parts of the original roots.

Conclusion

Therefore, \( p(p(x)) = 0 \) must have all real roots, because squaring purely imaginary numbers results in non-negative real numbers. We cannot determine the exact number of distinct real roots from the information given, but we can assert that they are all real.

Key concepts

Imaginary Roots

Imaginary roots of a quadratic equation occur when the equation has no real solutions. This happens when the discriminant, the part of the quadratic formula under the square root sign, is negative. The discriminant is calculated as \( b^2 - 4ac \). When it's less than zero, we can't take the square root of a negative number using real numbers, so we introduce an imaginary unit, \( i \), where \( i^2 = -1 \).

Thus, if a quadratic equation has real coefficients and its discriminant is negative, it will have two purely imaginary roots in the form \( bi \) and \( -bi \), with \( b eq 0 \) and \( i \) being the imaginary unit. The nature of imaginary roots is critical in understanding the behavior of quadratic equations and their compositions, as it can influence the nature and type of solutions when the function is composed with itself.

Real Coefficients

Real coefficients are the numerical factors of the terms in a polynomial equation that are not attached to any variable. In the quadratic equation \( ax^2 + bx + c = 0 \) all the coefficients \( a \) \( b \) and \( c \) are real numbers.

If a quadratic equation has real coefficients, any non-real roots must occur in conjugate pairs. This means that if the equation has one imaginary root \( a + bi \) the other root must be its conjugate \( a - bi \) ensuring that the coefficients remain real during the process of multiplication or function composition. This property is crucial when examining the roots of composed functions or higher-degree polynomials.

Function Composition

Function composition involves applying one function to the results of another. Mathematically, if you have two functions \( f(x) \) and \( g(x) \) the composition \( f(g(x)) \) represents the function \( f \) applied to the outcome of \( g(x) \) This is fundamental in advanced mathematics and has implications in various aspects of algebra and calculus.

In our context, composing the function \( p(x) \) with itself means we're looking at \( x \) values evaluated by \( p(x) \) and then re-evaluating the results through the same function \( p \) This concept is beautifully illustrated when we substitute \( x \) in \( p(x) \) with \( p(x) \) again to get \( p(p(x)) \)

Quartic Equation

A quartic equation is a polynomial equation of fourth degree, which can be written in the form \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) where the coefficients \( a \) \( b \) \( c \) \( d \) and \( e \) are real numbers, and \( a eq 0 \) Quartic equations can have up to 4 real roots, and the roots can be real, imaginary, or a mix of both.

In our exercise, the focus is on a quartic equation that arises from composing a quadratic function with purely imaginary roots. Squaring these imaginary roots results in a quartic equation that has all real coefficients. An important aspect of quartic equations worth mentioning is that they will always have an even number of non-real roots owing to the fact that they must occur in conjugate pairs when the coefficients are real.