Factoring Trinomials Where c Is Positive

Factorisation is a method to break arithmetic algebraic expressions into the product of their factors. When dealing with polynomials, the factors of the polynomials, when multiplied, produce the original polynomial. For example, the factors of x2 + 5x + 6 are (x + 2) (x + 3). We obtain the original polynomial by multiplying both x+2 and x+3.

Methods of factoring

The three most basic and most used ways of factoring a polynomial are

●      Greatest Common Factor (GCF)

●      Grouping Method

●      Factoring using algebraic identities.

A polynomial consisting of three terms is called a trinomial. Trinomials often have the form 

x2 + bx + c. At first glance, it may seem not easy to factor trinomials, but we can take advantage of some interesting algebraic mathematical patterns to factor even the most difficult-looking trinomials.

Illustration 1

Trinomials in the form x2 + bx + c can often be factored in as the product of two binomials. We have to remember that a binomial is a simple two-term polynomial. 

Let’s start by reviewing what happens if we multiply two binomials, such as (x + 2) and (x + 5).

(x + 2)(x + 5)

=x2 + 5x + 2x +10

Then we can combine terms 2x and 5x.

Therefore, the product comes out to be: x2 + 7x +10

Factoring is considered the reverse of multiplying, so if we go in reverse and factor the trinomial x2 + 7x + 10. The individual terms x2, 7x, and 10 shares no common factors. So we look at rewriting x2 + 7x + 10 as x2 + 5x + 2x + 10 here we have rewritten the middle term 7x as 5x + 2x.

Now, we can group pairs of factors :

Group the pairs and factor out the common factor x from the first pair and 2 from the second pair, and we get the result as: (x2 + 5x) + (2x + 10)

Factor each pair: x(x + 5) + 2(x + 5)

Then factor out the common factor x + 5: (x + 5)(x + 2)

Illustration 2

Polynomial to be factored is x2 + 5x + 6.

In this polynomial, the b part of the middle term 5x is 5, and the positive c term is 6. A chart will help us organise possibilities. On the left, list all possible factors of the c term, 6; on the right, we find the sums.

There are only two possible combinations of factors: 1 and 6 and 2 and 3. We can see that 2 + 3 = 5. So 2x + 3x = 5x, giving us the correct middle term.

We use values from the chart above and replace 5x with 2x + 3x.

x2 + 2x + 3x + 6

Grouping the pairs of terms,

(x2 + 2x) + (3x + 6)

Now we factor x out of the first pair of terms.

x(x + 2) + (3x + 6)

Now we factor 3 out of the second pair of terms.

x(x + 2) + 3(x + 2)

Now we factor out (x + 2).

(x + 2)(x + 3)

Therefore, the result comes out to be

(x + 2)(x + 3)

Conclusion

We can factorise a polynomial using many methods, most common among which are the greatest common factor method, the grouping method, and the algebraic identities method.

To factor a trinomial in the form x2 + bx + c, we find two integers, r, and s, whose product is c and whose sum is b.

We rewrite the trinomial as x2 + rx + sx + c and then use grouping and the distributive property to factor the polynomial. The resulting factors will be (x + r) and (x + s).

Here, ‘distribute’ means dividing something or giving a share or part of something. As per the distributive property, multiplying the sum of two or more addends by a number will provide us with the same result as multiplying each addend individually by the number and then adding the products together.

Factorisation of a polynomial can also be done by using algebraic identities. The most common identities used for factorisation of polynomials are:

●      (a + b)2 = a2 + 2ab + b2

●      (a – b)2 = a2 – 2ab + b2

●      a2 – b2= (a + b)(a – b)