Answered

Two polynomial expressions are added together to get $x+4 x^2-3+7 x^3$. Is the sum a polynomial?


AYes
BNo



Explain your reasoning.

Verified Answer :

  • A polynomial is an expression involving only addition, subtraction, multiplication, and non-negative integer exponents.
  • The expression \(E(x) = x + 4x^2 - 3 + 7x^3\) can be written in descending order as \(7x^3 + 4x^2 + x - 3\).
  • Each term features a constant coefficient and a non-negative integer exponent, which confirms it matches the standard definition of a polynomial.
  • Thus, the answer is (A) Yes. \(\boxed{\text{Yes}}\)

Explanation

  1. Definition and Analysis of Polynomials

A polynomial is an expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial is \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\), where \(n\) is a non-negative integer, and each \(a_i\) is a constant. We are given the expression \(E(x) = x + 4x^2 - 3 + 7x^3\). We need to determine if this expression meets the criteria to be classified as a polynomial.

  1. Evaluating the Expression and Conclusion

To compare \(E(x) = x + 4x^2 - 3 + 7x^3\) with the standard form, we rearrange the terms in descending order of their exponents:
\[E(x) = 7x^3 + 4x^2 + x - 3\]
Each term is of the form \(c \cdot x^n\), where \(c\) is a constant and \(n\) is a non-negative integer:

  • For \(7x^3\), the coefficient is \(7\) and the exponent is \(3\).
  • For \(4x^2\), the coefficient is \(4\) and the exponent is \(2\).
  • For \(x^1\), the coefficient is \(1\) and the exponent is \(1\).
  • For \(-3x^0\), the coefficient is \(-3\) and the exponent is \(0\).

Since all exponents (3, 2, 1, 0) are non-negative integers, the expression satisfies the definition. Furthermore, polynomials are closed under addition, meaning the sum of any two polynomials is always another polynomial. Thus, the sum is indeed a polynomial.

Examples

Polynomials are used in various real-life scenarios, such as engineering and physics, to model trajectories or physical phenomena. For example, if you throw a ball, its height over time can be represented by a quadratic polynomial \(h(t) = - \frac{1}{2}gt^2 + v_0 t + h_0\). Recognizing that expressions are polynomials allows scientists and engineers to use calculus and algebraic methods to analyze motion, predict outcomes, and optimize designs efficiently.

Related Questions