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Chapter 2: Problem 76

Suppose \(q\) is a polynomial of degree 5 such that \(q(1)=-3\). Define \(p\) by $$ p(x)=x^{6}+q(x) $$ Explain why \(p\) has at least two zeros.

Short Answer

Expert verified
In conclusion, the polynomial \(p(x) = x^6 + q(x)\) has at least two zeros due to the conditions given in the problem and the nature of its degree. Since p(x) is a polynomial of even degree (degree 6) and p(1) 鈮� 0, p(x) must have a different zero than x = 1. Furthermore, since p(x) has opposite behavior as x goes to 卤鈭�, there must be a zero between the two, and p(x) must cross the x-axis at least twice, meaning it has at least two zeros.

Step by step solution

01

Understand the polynomial p(x)

Let's analyze the polynomial p(x) = x^6 + q(x). It is the sum of a polynomial of degree 6, x^6, and the polynomial q(x) of degree 5. The resulting polynomial p(x) will have a degree of 6, as the sum keeps the highest degree.
02

Analyze the polynomial q(x)

We know that q(x) is a polynomial of degree 5 and q(1) = -3. Remember that the degree of a polynomial tells us the maximum number of zeros it can have.
03

Identify the zeros of q(x)

q(x) is a polynomial of odd degree; hence, it must have at least one zero, but we are not given any specific zeros of q(x). We do know that q(1) = -3, which implies that x = 1 is not a zero of q(x).
04

Calculate p(1)

Now, let us calculate the value of p(1) using the given definition of p(x): \[ p(1) = (1)^6 + q(1) = 1 - 3 = -2 \]Since p(1) is not equal to 0, x = 1 is not a zero of the polynomial p(x).
05

Observe the behavior of p(x) as x approaches 卤鈭�

Since p(x) is a polynomial of even degree (degree 6), and its leading coefficient is positive, the polynomial will have the same behavior as x goes to 卤鈭�. As x 鈫� -鈭�, p(x) 鈫� +鈭�, and as x 鈫� +鈭�, p(x) 鈫� +鈭�.
06

Conclude that p(x) has at least two zeros

Since p(x) is a polynomial of even degree (degree 6) and p(1) 鈮� 0, p(x) must have a different zero than x = 1. Furthermore, since p(x) has opposite behavior as x goes to 卤鈭�, there must be a zero between the two, and p(x) must cross the x-axis at least twice, meaning it has at least two zeros. In conclusion, the polynomial p(x) = x^6 + q(x) has at least two zeros due to the conditions given in the problem and the nature of its degree.

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