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Chapter 2: Problem 40

Suppose \(p\) and \(q\) are polynomials of degree 3 such that \(p(1)=q(1), p(2)=q(2), p(3)=q(3),\) and \(p(4)=q(4) .\) Explain why \(p=q\).

Short Answer

Expert verified
Let \(d(x) = p(x) - q(x)\), where \(d(x)\) represents the difference between the polynomials \(p(x)\) and \(q(x)\). Since \(p(x)\) and \(q(x)\) are 3rd degree polynomials, their difference, \(d(x)\), will also be a 3rd degree polynomial or lower. We are given that \(d(1) = d(2) = d(3) = d(4) = 0\), so \(d(x)\) has roots at \(x = 1, 2, 3, 4\). Since a non-zero polynomial of degree n can have at most n roots, and \(d(x)\) has 4 roots, it must be the zero polynomial. Therefore, \(d(x) = p(x) - q(x) = 0\), which implies \(p(x) = q(x)\) for all \(x\). Hence, polynomials \(p\) and \(q\) are the same.

Step by step solution

01

Find the difference between the polynomials

Let \(d(x) = p(x) - q(x)\), where \(d(x)\) represents the difference between the polynomials \(p(x)\) and \(q(x)\). Now we have a new polynomial \(d(x)\).
02

Calculate the degree of the difference polynomial

Since both \(p(x)\) and \(q(x)\) are 3rd degree polynomials, their difference, \(d(x)\), will also be a 3rd degree polynomial or lower if some terms cancel out.
03

Evaluate the difference polynomial at given points

We are given that \(p(1)=q(1)\), \(p(2)=q(2)\), \(p(3)=q(3)\), and \(p(4)=q(4)\). Substitute these values into the difference polynomial: 1. \(d(1) = p(1) - q(1) = 0\) 2. \(d(2) = p(2) - q(2) = 0\) 3. \(d(3) = p(3) - q(3) = 0\) 4. \(d(4) = p(4) - q(4) = 0\) So, \(d(x)\) has roots at \(x = 1, 2, 3, 4\).
04

Determine if the difference polynomial is the zero polynomial

By the Fundamental Theorem of Algebra, a non-zero polynomial of degree n can have at most n roots. In our case, \(d(x)\) is a polynomial of degree 3 (or lower) and has 4 roots, which means it must be the zero polynomial.
05

Conclude that the two given polynomials are equal

Since \(d(x) = p(x) - q(x) = 0\), we can conclude that \(p(x) = q(x)\) for all \(x\). Hence, the polynomials \(p\) and \(q\) are the same.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Polynomials
When dealing with polynomials, finding the difference between two of them involves subtracting their respective terms. In the exercise, we defined a new polynomial as the difference:
  • Let \( d(x) = p(x) - q(x) \)
This new polynomial \( d(x) \) reflects the variation between \( p(x) \) and \( q(x) \). If \( d(x)=0\) for all values of \(x\), then \( p(x)\) and \(q(x)\) are equivalent throughout their domain. It implies that every corresponding term in \(p(x)\) and \(q(x)\) must cancel each other out.
Thus, the difference of polynomials helps us see if two polynomials are equal by checking whether their difference is a zero polynomial.
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra is a central principle in algebra. It states that a non-zero polynomial of degree \( n \) has exactly \( n \) roots in the complex number system, counting multiplicities.
  • In our exercise, \(d(x)\) is a polynomial formed by subtracting \( q(x) \) from \( p(x) \).
  • The degree of \(d(x)\) is at most 3 because that's the degree of both \( p(x) \) and \( q(x) \).
If \(d(x)\) has more roots than its degree, it must be the zero polynomial. In this case, \( d(x) \) has four roots (at \( x = 1, 2, 3, 4 \)), which exceeds its degree of 3 if non-zero. The theorem helps confirm that \( d(x) \) is zero everywhere, hence proving \( p(x) = q(x) \).
Polynomial Roots
In the context of polynomials, 'roots' are the values of \(x\) that make the polynomial equal zero. For \(d(x) = p(x) - q(x)\) in this exercise, we want to find the values where \(d(x) = 0\). According to the given problem:
  • \( d(1) = 0 \)
  • \( d(2) = 0 \)
  • \( d(3) = 0 \)
  • \( d(4) = 0 \)
This means that the polynomial \(d(x)\) has four roots at \( x = 1, 2, 3,\) and \( 4 \). Adding more roots than the degree theoretically allows means \(d(x)\) is zero since it cannot possess more distinct roots than its own degree. By finding these roots, we affirm that \(p(x)\) and \(q(x)\) must be equivalent.
Degree of Polynomial
The degree of a polynomial is the highest power of \(x\) with a non-zero coefficient in the polynomial expression. It tells us the maximum number of roots that the polynomial can have. For polynomials \(p(x)\) and \(q(x)\) in our exercise:
  • Both are polynomials of degree 3.
  • Their difference \(d(x) = p(x) - q(x)\) would be at most a polynomial of degree 3, unless all terms cancel.
If a polynomial like \(d(x)\) has more roots than its degree allows and must have degree 3 or less, this implies it must be the zero polynomial. In other words, polynomial expressions like these reveal their parity when analyzed by the degree. In this specific exercise, it is leverage the degree of polynomials to moralize that \(p(x) = q(x)\).

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Most popular questions from this chapter

Let \(p\) be the polynomial defined by $$p(x)=x^{6}-87 x^{4}-92 x+2$$. (a)Use a computer or calculator to sketch a graph of \(p\) on the interval [-5,5] . (b) Is \(p(x)\) positive or negative for \(x\) near \(\infty ?\) (c) Is \(p(x)\) positive or negative for \(x\) near \(-\infty ?\) (d) Explain why the graph from part (a) does not accurately show the behavior of \(p(x)\) for large values of \(x\).

Let \(p\) be the polynomial defined by $$p(x)=x^{6}-87 x^{4}-92 x+2$$. (a)Evaluate \(p(-2), p(-1), p(0),\) and \(p(1)\). (b) Explain why the results from part (a) imply that \(p\) has a zero in the interval (-2,-1) and \(p\) has a zero in the interval (0,1) . (c) Show that \(p\) has at least four zeros in the interval [-10,10] . [Hint: We already know from part ( \(b\) ) that \(p\) has at least two zeros is the interval [-10,10] . You can show the existence of other zeros by finding integers \(n\) such that one of the numbers \(p(n)\), \(p(n+1)\) is positive and the other is negative.]

Suppose \(s(x)=4 x^{3}-2\) (a) Show that the point (1,2) is on the graph of \(s\). (b) Give an estimate for the slope of a line containing (1,2) and a point on the graph of \(s\) very close to (1,2) [Hint: Use the result of Exercise \(18 .]\)

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{6}-4 x^{2}+5}{x^{2}-3 x+1} $$

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{9 x+5}{x^{2}-x-6} $$
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