b) 3p2 5pq 2q2 – – Use decomposition. Since the 3rd term in the trinomial is negative, the operations in the binomial factors will be addition and subtraction. The two binomials will have the form: (?p + ?q)(?p ?q) – The product of the coefficients of and is: 3(–2) = p2 q2 –6 Write as a sum of two terms whose coefficients have a product of –5pq –6. Factors of Sum of Factors –6 1, 1 6 = –6 – –5 6 + 6 = 5 –1, –1 2, 2 3 = –3 – –1 3 + 3 = 1 –2, –2 The two coefficients are 1 and so write the trinomial –6, 3p2 5pq 2q2 as 3p2 + 6pq 2q2. pq – – – – Remove a common factor from the 1st pair of terms, and from the 2nd pair of terms: 3p2 + 6pq 2q2 = + 2q(3p + pq – – p(3p q) – q) Remove the common binomial factor. So, 3p2 5pq 2q2 = (3p + 2q) q)(p – – – To verify, multiply: (3p + 2q) = 3p(p 2q) + 2q) q)(p q(p – – – = 3p2 6pq + 2q2 – pq – = 3p2 5pq 2q2 – – Since this trinomial is the same as the original trinomial, the factors are correct.
b) 3p2 – 5pq – 2q2 Use decomposition. Since the 3rd term in the trinomial is negative, the operations in the binomial factors will be addition and subtraction. The two binomials will have the form: (?p + ?q)(?p – ?q) The product of the coefficients of p2 and q2 is: 3(–2) = –6 Write –5pq as a sum of two terms whose coefficients have a product of –6. Factors of –6 Sum of Factors 1, –6 1 – 6 = –5 –1, 6 –1 + 6 = 5 2, –3 2 – 3 = –1 –2, 3 –2 + 3 = 1 The two coefficients are 1 and –6, so write the trinomial 3p2 – 5pq – 2q2 as 3p2 + pq – 6pq – 2q2. Remove a common factor from the 1st pair of terms, and from the 2nd pair of terms: 3p2 + pq – 6pq – 2q2 = p(3p + q) – 2q(3p + q) Remove the common binomial factor. So, 3p2 – 5pq – 2q2 = (3p + q)(p – 2q) To verify, multiply: (3p + q)(p – 2q) = 3p(p – 2q) + q(p – 2q) = 3p2 – 6pq + pq – 2q2 = 3p2 – 5pq – 2q2 Since this trinomial is the same as the original trinomial, the factors are correct.