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Chapter 2: Problem 31

Give an example of two polynomials of degree 4 whose sum has degree 3 .

Short Answer

Expert verified
An example of two polynomials of degree 4 whose sum has degree 3 is P(x) = \(x^4 + x^3 + x^2 + x + 1\) and Q(x) = \(-x^4 + 2x^3 + 3x^2 - x + 4\). When summed, R(x) = P(x) + Q(x) = \(3x^3 + 4x^2 + 5\), which is a polynomial of degree 3.

Step by step solution

01

Understand what makes a polynomial of degree 4

A polynomial of degree 4 is a polynomial with the highest exponent of the variable being 4. The general form of a polynomial of degree 4 is: \[ax^4 + bx^3 + cx^2 + dx + e\] The coefficients a, b, c, d, and e can be any real numbers, with 'a' being non-zero.
02

Consider two polynomials of degree 4

Let's consider two polynomials P(x) and Q(x) of degree 4: P(x) = \(a_1x^4 + a_2x^3 + a_3x^2 + a_4x + a_5\) Q(x) = \(b_1x^4 + b_2x^3 + b_3x^2 + b_4x + b_5\) Where \(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\), \(b_1\), \(b_2\), \(b_3\), \(b_4\), and \(b_5\) are real numbers and \(a_1\) and \(b_1\) are non-zero.
03

Find the sum of P(x) and Q(x)

Let's calculate the sum of these two polynomials, R(x): R(x) = P(x) + Q(x) = \((a_1 + b_1)x^4 + (a_2 + b_2)x^3 + (a_3 + b_3)x^2 + (a_4 + b_4)x + (a_5 + b_5)\)
04

Make the sum polynomial have degree 3

For the sum polynomial R(x) to have degree 3, we need to eliminate the term with exponent 4. To do so, the coefficient of \(x^4\) in R(x) must be equal to 0: \(a_1 + b_1 = 0\) This means \(b_1 = -a_1\). So, we will now have a sum polynomial of degree 3: R(x) = \((a_2 + b_2) x^3 + (a_3 + b_3) x^2 + (a_4 + b_4) x + (a_5 + b_5)\)
05

Determine the coefficients of the two polynomials

We can now choose the coefficients for the polynomials P(x) and Q(x), ensuring the sum polynomial R(x) remains degree 3: P(x) = \(x^4 + x^3 + x^2 + x + 1\) Q(x) = \(-x^4 + 2x^3 + 3x^2 - x + 4\) Notice that the coefficients of \(x^4\) in P(x) and Q(x) are 1 and -1 respectively, so their sum will be zero. The sum of P(x) and Q(x) will be: R(x) = \((1 + 2) x^3 + (1 + 3) x^2 + (1 - 1) x + (1 + 4)\) R(x) = \(3x^3 + 4x^2 + 5\) The polynomials P(x) and Q(x) are both degree 4 polynomials, and their sum R(x) is a degree 3 polynomial, as required.

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Most popular questions from this chapter

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s(x))^{2} $$

Suppose \(p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n},\) where \(a_{0}, a_{1}, \ldots, a_{n}\) are integers. Suppose \(M\) and \(N\) are nonzero integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\). Show that \(a_{0} / M\) and \(a_{n} / N\) are integers. [Thus to find rational zeros of a polynomial with integer coefficients, we need only look at fractions whose numerator is a divisor of the constant term and whose denominator is a divisor of the coefficient of highest degree. This result is called the Rational Zeros Theorem or the Rational Roots Theorem.]

Suppose $$r(x)=\frac{x+1}{x^{2}+3} \quad \text { and } \quad s(x)=\frac{x+2}{x^{2}+5}$$ What is the domain of \(r ?\)

Suppose \(p\) and \(q\) are polynomials and the horizonal axis is an asymptote of the graph of \(\frac{p}{q}\). Explain why $$ \operatorname{deg} p<\operatorname{deg} q $$

$$ \text { Suppose } p(x)=2 x^{6}+3 x^{5}+5 $$ (a) Show that if \(\frac{M}{N}\) is a zero of \(p\), then $$ 2 M^{6}+3 M^{5} N+5 N^{6}=0 $$ (b) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(5 / M\) and \(2 / N\) are integers. (c) Show that the only possible rational zeros of \(p\) $$ \text { are }-5,-1,-\frac{1}{2}, \text { and }-\frac{5}{2} \text { . } $$ (d) Show that no rational number is a zero of \(p\).
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