Factoring Trinomials (a > 1)

Concept: AC Method (Factoring when a > 1)

1 Write the trinomial in standard form: \(ax^2 + bx + c\)

2 Find the product \(A = a \cdot c\). Find two integers \(m, n\) such that \(m \cdot n = A\) and \(m + n = b\).

3 Rewrite the middle term: split \(bx\) into \(mx + nx\).

4 Factor by grouping the four terms into two pairs, then factor out the GCF from each pair.

5 Factor out the common binomial. Always check by expanding (FOIL).

★ If \(a \neq 1\), always check first whether a GCF can be factored out before applying the AC method.

Factor \(2x^2 + 7x + 3\) completely.

① \(a=2,\ b=7,\ c=3\) → \(A = 2 \times 3 = 6\)

② Find \(m,n\): \(m \cdot n = 6,\ m+n = 7\) → \(m=6,\ n=1\)

③ Rewrite: \(2x^2 + 6x + x + 3\)

④ Group: \(2x(x+3) + 1(x+3)\)

⑤ Factor: \(\boxed{(2x+1)(x+3)}\)

✓ Check: \((2x+1)(x+3)=2x^2+6x+x+3=2x^2+7x+3\) ✔