Question 37 of 49beginner✨ CreateShort Answer2 marks

Write a polynomial of degree 3 in the variable x, in which the coefficient of the x2x^{2} term is -7.

Correct Answer

A polynomial of degree 3 in the variable xx means that the highest power of xx in the expression must be 3. As provided in the context, such polynomials are called cubic polynomials. For example, an expression like ax3+bx2+cx+dax^3 + bx^2 + cx + d has degree 3 if aa is not zero.

The question requires the coefficient of the x2x^2 term to be −7-7. In the expression, this means the number multiplying x2x^2 must be −7-7. The other terms can have any coefficients, provided the x3x^3 term exists to maintain degree 3. Thus, a valid polynomial is x3−7x2+2x+5x^3 - 7x^2 + 2x + 5. Another correct example could simply be x3−7x2x^3 - 7x^2.

Exercise: END-OF-CHAPTER EXERCISES | Q: 1 | (Page 21)
For More Understanding

Explanation

The question requires constructing a cubic polynomial with a specific coefficient. The context defines the degree of a polynomial as the highest power of the variable and identifies degree 3 polynomials as cubic polynomials. It also explains how to identify coefficients. By ensuring the highest power is 3 and the coefficient of the squared term is -7, we can construct multiple valid answers.

Solution Steps

  1. Step 1: Identify that a polynomial of degree 3 must have the highest power of the variable as 3 (e.g., an x3x^3 term must be present with a non-zero coefficient).

  2. Step 2: Set the coefficient of the x2x^2 term to -7, as required by the question.

  3. Step 3: Choose arbitrary values for the remaining coefficients to form a complete polynomial, such as x3−7x2+2x+5x^3 - 7x^2 + 2x + 5.

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