2. Unit 3: Algebraic Skills
  3. Lesson 2: Factoring — Common and Trinomials

Factoring \(x^2 +bx+c\)


Slide Notes

Glossary

All Slides

Navigation

Four Types of Factoring

 

Factoring \(x^2+bx+c\)

Recall

Factoring and expanding are opposites.

 

Factoring \(x^2+bx+c\) Continued 

Let's expand the following: 

\((x+4)(x+5)\)

 

 

Steps to Factor \(x^2+bx+c\)

Before we start factoring, let's try another expanding question. 

 

 

 

 

Example 3

Factor \(x^2+12x+32\).

 

Paused Finished
Slide /

Check Your Understanding 4


Part 1: null
How Did I Do?Try Another

Example 4

Factor \(x^2-x-20\).

Solution

We require a pair of integers that

  • multiply to \(-20\), and
  • add to \(-1\).

Testing pairs of integers with product \(-20\):

Number 1 Number 2 Sum Correct?
\(1\) \(-20\) \(-19\) ✘
\(2\) \(-10\) \(-8\) ✘
\(4\) \(-5\) \(-1\) ✔

\(x^2-x-20 =(x+4)(x-5)\)

Hints to Help Determine the Two Numbers

  • Start with the product of the pair of integers.
  • If the product is negative, one number will be positive and the other will be negative. The sign of the sum of the two numbers indicates the sign of the larger of the two numbers.
  • If the product is positive, either both numbers are positive or both numbers are negative. The sign of the sum of the two numbers indicates whether the two numbers are positive or negative.

Check Your Understanding 5


Factor \($fullExp(details...)\).

Enter factors in the form "\((ax+b)(cx+d)\)".

Factored form: There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

null

How Did I Do?Try Another

Example 5

Change \(y=3x^2+6x-45\) to factored form, then determine the zeros.

Solution

It may seem like this question is misplaced since this lesson part is about factoring \(x^2+bx+c\) so the coefficient of \(x^2\) is supposed to be \(1\). However, this question involves two steps:

  1. First common factor.
  2. Then factor the resulting trinomial which will be of the form \(x^2+bx+c\).

\(y\)

 \(=3x^2+6x-45\)

Factor out the common factor of \(3\)

 

 \(=3(x^2+2x-15)\)

Think of two numbers that add to \(2\) and multiply to \(-15\)

 

 \(=3(x+5)(x-3)\)

 

Thus, the factored form equation is \(y=3(x+5)(x-3)\).

  • \(y = 3(x+5)(x-3)\) has zeros \(-5\) and \(3\).
  • \(y = 3(x+5)(x-3)\) describes the same parabola as \(y=3x^2+6x-45\).
  • Therefore, \(y=3x^2+6x-45\) has zeros \(-5\) and \(3\).

Check Your Understanding 6


Part 2: null
How Did I Do?Try Another

Example 6

Factor \(x^2+4x+5\).

Solution

We require a pair of integers that have

  • product \(5\), and 
  • sum \(4\).

Testing pairs of integers with product \(5\):

Number 1 Number 2 Addition Correct?
\(1\) \(5\) \(6\) ✘
\(-1\) \(-5\) \(-6\) ✘

There are no pairs of integers that multiply to \(5\) and add to \(4\). This expression is not factorable.

Let's take a look at the corresponding parabola, \(y=x^2+4x+5\), to see why this is the case.

The parabola doesn't cross the x-axis and opens upwards.

If \(y=x^2+4x+5\) was factorable, we would be able to determine the zeros from its factored form, but observing this parabola we can see it does not have any zeros. As a result, it does not have a factored form.

 

\(y=ax^2+bx+c\) cannot be factored using integers when the associated parabola has

  • no zeros, or
  • irrational zeros.

 

In a future lesson part, we will consider the factored form for parabolas that have rational but non-integer zeros.