Introduction to Linear Polynomials | Exercise 2.1

Question 3

What are the coefficients of x2x^2 and x3x^3 in the polynomial x4−3x3+6x2−2x+7x^4 - 3x^3 + 6x^2 - 2x + 7?

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Solution
Understand the Question
  • The coefficient of a variable in a term is the numerical constant multiplying that variable (including its positive or negative sign).
  • Given the polynomial P(x)=x4−3x3+6x2−2x+7P(x) = x^4 - 3x^3 + 6x^2 - 2x + 7, we locate the terms containing x2x^2 and x3x^3 to identify their respective numerical factors.

Step 1 · Identify the Polynomial

Given polynomial P(x)=x4−3x3+6x2−2x+7P(x) = x^4 - 3x^3 + 6x^2 - 2x + 7

Diagram 1

Step 2 · Find the Coefficient of x3x^3

The term containing x3x^3 is −3x3-3x^3. The coefficient of x3 is −3\text{The coefficient of } x^3 \text{ is } -3

Step 3 · Find the Coefficient of x2x^2

The term containing x2x^2 is 6x26x^2. The coefficient of x2 is 6\text{The coefficient of } x^2 \text{ is } 6

Answer

The coefficient of x2x^2 is 66 and the coefficient of x3x^3 is −3-3.

Common Mistakes
  • Ignoring the Negative Sign: Writing the coefficient of x3x^3 as 33 instead of −3-3. Always include the sign attached to the term.
  • Writing the Whole Term: Writing 6x26x^2 or −3x3-3x^3 as the coefficient instead of just the numerical factor (66 and −3-3).

More questions in Exercise 2.1

Q1

Find the degrees of the following polynomials:

(i) 2x2−5x+32x^2 - 5x + 3

(ii) y3+2y−1y^3 + 2y - 1

(iii) −9-9

(iv) 4z−34z - 3

Q2

Write polynomials of degrees 1, 2 and 3.

Q3

What are the coefficients of x2x^2 and x3x^3 in the polynomial x4−3x3+6x2−2x+7x^4 - 3x^3 + 6x^2 - 2x + 7?

Q4

What is the coefficient of zz in the polynomial 4z3+5z2−114z^3 + 5z^2 - 11?

Q5

What is the constant term of the polynomial 9x3+5x2−8x−109x^3 + 5x^2 - 8x - 10?

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