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Chapter 2: Problem 34

Show that if \(p\) and \(q\) are nonzero polynomials with \(\operatorname{deg} p<\operatorname{deg} q,\) then \(\operatorname{deg}(p+q)=\operatorname{deg} q\).

Short Answer

Expert verified
Given two nonzero polynomials p(x) and q(x) with deg p < deg q, we can express them as: p(x) = \(a_m x^m + ... + a_1 x + a_0\), where \(a_m\) is nonzero q(x) = \(b_n x^n + ... + b_1 x + b_0\), where \(b_n\) is nonzero Adding the polynomials gives p(x) + q(x) = \(b_n x^n + (b_{n-1} + a_m) x^{n-1} + ...+ (a_1 + b_1) x + (a_0 + b_0)\). Since there is no term in p(x) that can cancel the highest-degree term in q(x) with a degree of n, we can conclude that deg(p+q) = n, which means deg(p+q) = deg q.

Step by step solution

01

Define the polynomials p and q

Let p(x) and q(x) be two nonzero polynomials with variable x. By definition, the degree of a polynomial is the highest power of the variable in the polynomial. We are given that deg p < deg q. Let deg p = m and deg q = n, where m and n are non-negative integers, and m < n. Now, express p(x) and q(x) in terms of their coefficients and variable. p(x) = \(a_m x^m + a_{m-1} x^{m-1} + ... + a_1 x + a_0\), where \(a_m\) is nonzero q(x) = \(b_n x^n + b_{n-1} x^{n-1} + ... + b_1 x + b_0\), where \(b_n\) is nonzero
02

Find the sum of the polynomials p and q

Next, we need to find the sum of p(x) and q(x). p(x) + q(x) = \((a_m x^m + ... + a_1 x + a_0) + (b_n x^n + ... + b_1 x + b_0)\)
03

Combine like terms and simplify

Now, we will combine like terms in the sum of p(x) and q(x), which will give us the coefficients of the resulting polynomial (p+q). Since deg p = m < n = deg q, the highest-degree term of p(x) has a smaller exponent than that of q(x), so, when we add the polynomials, there will be no term in p(x) that can cancel the highest-degree term in q(x), which has a degree of n. Thus, p(x) + q(x) = \(b_n x^n + (b_{n-1} + a_m) x^{n-1} + ...+ (a_1 + b_1) x + (a_0 + b_0)\)
04

Determine the degree of (p+q)

Since the coefficients of p(x) + q(x) now include the coefficient corresponding to the highest-degree term of q(x) (with degree n), and there is no term in p(x) with a degree greater than or equal to n, the sum p(x) + q(x) includes the highest-degree term, \(b_n x^n\), which is nonzero. Therefore, we can conclude that deg(p+q) = n because the highest-degree term in p(x) + q(x) has a degree of n.
05

Conclusion: Deg(p+q) equals deg q

Finally, we have shown that the degree of the sum of the two polynomials (p+q) will have the same degree as q. In other words, deg(p+q) = deg q.

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