Polynomials and Factorisation

(x + 2)(x + 3)

Step 1: We need two numbers that multiply to 6 and add to 5. Step 2: The pairs that multiply to 6 are: (1,6) and (2,3). Step 3: Check which pair adds to 5: 1 + 6 = 7 (no), 2 + 3 = 5 (yes). Step 4: Therefore, x² + 5x + 6 = (x + 2)(x + 3). Step 5: Verify: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓

9x² + 12x + 4

Step 1: Identify a = 3x and b = 2 in the identity (a + b)² = a² + 2ab + b². Step 2: Calculate a² = (3x)² = 9x². Step 3: Calculate 2ab = 2(3x)(2) = 12x. Step 4: Calculate b² = (2)² = 4. Step 5: Combine: (3x + 2)² = 9x² + 12x + 4.

Trinomial

Step 1: Count the number of terms in the polynomial 7x³ - 4x + 1. Step 2: The terms are: 7x³, -4x, and 1. Step 3: There are exactly 3 terms. Step 4: A polynomial with 3 terms is called a trinomial. Remember: 1 term = monomial, 2 terms = binomial, 3 terms = trinomial.

-1

Step 1: By the Remainder Theorem, when p(x) is divided by (x - a), the remainder is p(a). Step 2: Here, we're dividing by (x - 1), so a = 1. Step 3: Calculate p(1): p(1) = (1)³ - 2(1)² + (1) - 1. Step 4: Simplify: p(1) = 1 - 2 + 1 - 1 = -1. Step 5: Therefore, the remainder is -1.