To factor trinomials ax2 + bx + c when a ≠ 1, we can use the AC method (Splitting the Middle Term) i.e., multiply a and c, find factors summing to b, rewrite, group, and factor.
This methods is explained in this article in detail including example.
What is Trinomial?
A trinomial is a type of polynomial that consists of exactly three terms. In algebra, a polynomial is an expression made up of variables (also called indeterminates), coefficients, and exponents that are combined using addition, subtraction, and multiplication. A trinomial can be represented in the form:
axn + bxm + cxp
Where a, b, and c are coefficients, and n, m, and p are non-negative integers representing the exponents. The variables x can also be different.
Most common trinomial which is commonly studied by students is quadratic:
ax2 + bx + c = 0
Here, we will discuss the method to factorize this trinomial.
Steps to Factor Trinomials when a is not equals to 1
Steps to factorize a trinomial with second degree when a is not equal to 1 are:
Step 1: Multiply a and c: Calculate the product ac.
Step 2: Find Two Numbers that Multiply to ac and Sum up to b: Identify two numbers m and n such that m × n = ac and m + n = b.
Step 3: Rewrite the Middle Term Using m and n: Rewrite the trinomial ax2 + bx + c as ax2 + mx + nx + c.
Step 4: Factor by Grouping: Group the terms into two pairs and factor out the greatest common factor (GCF) from each pair.
Step 5: Factor Out the Common Binomial Factor: After factoring each pair, you should have a common binomial factor which you can factor out.
Let's consider an example for better understanding.
Example for Trinomial Factorization
Let's factor 6x2 + 11x + 4.
Multiply a and c: ac = 6 × 4 = 24
Find Two Numbers that Multiply to 24 and Add to 11: The numbers 8 and 3 satisfy the conditions since 8 × 3 = 24 and 8 + 3 = 11.
Rewrite the Middle Term: 6x2 + 11x + 4 = 6x2 + 8x + 3x + 4
Factor by Grouping: Group the terms:(6x2 + 8x) + (3x + 4)
Factor out the GCF from each group: 2x(3x + 4) + 1(3x + 4)
Factor Out the Common Binomial Factor: (2x + 1)(3x + 4)
So, the factored form of 6x2 + 11x + 4 is (2x + 1)(3x + 4)