Polynomials — Important Questions

Quadratic

Step 1: Find the degree of the polynomial 2x² + 5x - 3. Step 2: The highest power of x is 2 (from the term 2x²). Step 3: A polynomial of degree 2 is called a quadratic polynomial. Step 4: Therefore, 2x² + 5x - 3 is a quadratic polynomial. Remember: degree 1 = linear, degree 2 = quadratic, degree 3 = cubic.

3

Step 1: To find the zero, set p(x) = 0. So we need to solve 3x - 9 = 0. Step 2: Add 9 to both sides: 3x = 9. Step 3: Divide both sides by 3: x = 3. Step 4: Verify: p(3) = 3(3) - 9 = 9 - 9 = 0 ✓. Therefore, 3 is the zero of the polynomial.

4x² + 12x + 9

Step 1: Compare (2x + 3)² with (x + y)² to identify x = 2x and y = 3. Step 2: Apply the identity (x + y)² = x² + 2xy + y². Step 3: Substitute: (2x)² + 2(2x)(3) + (3)². Step 4: Calculate each term: (2x)² = 4x², 2(2x)(3) = 12x, (3)² = 9. Step 5: Therefore, (2x + 3)² = 4x² + 12x + 9.

4

Step 1: Identify each separate term in the polynomial x⁴ - 2x³ + x - 5. Step 2: The terms are: x⁴ (first term), -2x³ (second term), x (third term), and -5 (fourth term). Step 3: Count the total number of terms. Step 4: There are 4 terms in total. Remember: terms are separated by + or - signs.