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## Digital SAT Math

### Course: Digital SAT Math > Unit 12

Lesson 1: Factoring quadratic and polynomial expressions: advanced# Factoring quadratic and polynomial expressions | Lesson

A guide to factoring quadratic and polynomial expressions on the digital SAT

## What are factoring quadratic and polynomial expressions questions?

Factoring quadratic and polynomial expressions questions ask you to rewrite polynomials in their equivalent, factored form.

For example, ${x}^{2}+3x+2$ can be rewritten as $(x+1)(x+2)$ , and ${x}^{2}-9$ can be rewritten as $(x-3)(x+3)$ .

In this lesson, we'll learn to:

- Factor polynomial expressions
- Use knowledge of factoring to evaluate expressions

**You can learn anything. Let's do this!**

## How do I factor quadratic expressions?

### Factoring quadratic expressions

**Note:**We cover how to factor quadratic expressions in the form

**Solving quadratic equations**lesson.

Quadratic expressions that have an ${x}^{2}$ -coefficient that is not $1$ are more difficult to factor. We should try to factor out any common factors if possible.

For example, in the expression $3{x}^{2}+12x-15$ , we can factor out a $3$ first and then factor the quadratic expression ${x}^{2}+4x-5$ .

To factor a quadratic expression in the form ${x}^{2}+bx+c$ :

- Find two numbers with a product equal to
and a sum equal to$c$ .$b$ - The two factors of the expression are each the sum of
and one of the numbers from Step 1.$x$

**Example:**Factor

When we can't factor out an integer, we can use a technique called

**factoring by grouping**.### Factoring quadratics by grouping

If you'd like to review this technique, we recommend watching this video before proceeding!

### How do we factor by grouping?

The first step of factoring $a{x}^{2}+bx+c$ is familiar: we're looking for two integers with a product equal to $ac$ and a sum equal to $b$ .

For example, let's factor $2{x}^{2}+9x-5$ .

We are looking for two numbers that meet the following criteria:

- Their product is equal to
.$(2)(-5)=-10$ - Their sum is equal to
.$9$

The numbers $10$ and $-1$ work:

$(10)(-1)=-10$ $10+(-1)=9$

These two numbers alone do $x$ -term of the expression. We can rewrite $9x$ as $10x-x$ :

*not*give us the factors. Instead, they tell us how to split up theNext, we

*group*the terms into two pairs:Notice that we can factor each pair of terms:

When we do this, we see that both of the initial pairs contain the factor $x+5$ . This means we can factor out the $x+5$ from the overall expression:

Therefore, $2{x}^{2}+9x-5=(x+5)(2x-1)$ .

**Note:**Factoring by grouping can be difficult, and there are often other strategies that can get you to the answer on test day. For example, you might be able to simply test multiple choice options by using FOIL, or plug in simple values for

To factor a quadratic expression in the form $a{x}^{2}+bx+c$ :

- Factor out any integers if possible. If this results in the product of an integer and a quadratic expression in the form
, follow the steps for factoring${x}^{2}+bx+c$ shown above.${x}^{2}+bx+c$ - Find two numbers with a product equal to
and a sum equal to$ac$ .$b$ - Use the two numbers from Step 2 to split
into two$bx$ -terms.$x$ - Group the resulting expression into two pairs of terms: one pair should have an
-term and an${x}^{2}$ -term, and the other pair should have an$x$ -term and a constant term.$x$ - Factor out an expression containing
from the pair with an$x$ -term and an${x}^{2}$ -term. Factor out a constant from the pair with an$x$ -term and a constant term. These two pairs should now share a binomial factor.$x$ - The shared binomial factor is one factor of the quadratic expression. The expression containing
and the constant factored out in Step 5 combine to form the other factor of the quadratic expression.$x$

**Example:**Factor

### Try it!

## How do I use special factoring?

### Factor difference of squares: leading coefficient $\ne 1$

### Special factoring

Special factoring rules are shortcuts for factoring polynomials with specific combinations of terms. Many students find that when they recognize opportunities to apply special factoring rules, they save valuable time on test day.

Anytime we see multiple perfect square terms in a polynomial expression, e.g., $9$ or $4{x}^{2}$ , check the other terms to see whether the expression satisfies the criteria for special factoring.

**Square of sum:**

**Square of difference:**

**Difference of squares:**

To use special factoring to factor a polynomial expression:

- Factor out any common factors if possible.
- Recognize that one or more terms in the expression are perfect squares.
- Confirm that all of the terms in the expression satisfy the criteria for special factoring.
- Apply the appropriate special factoring rule.

#### Let's look at some examples!

Factor $4{x}^{2}+12x+9$ .

Factor $9{y}^{4}-25{x}^{2}$ .

### Try it!

## Your turn!

## Things to remember

Square of sum: ${a}^{2}+2ab+{b}^{2}=(a+b{)}^{2}$

Square of difference: ${a}^{2}-2ab+{b}^{2}=(a-b{)}^{2}$

Difference of squares: ${a}^{2}-{b}^{2}=(a+b)(a-b)$

## Want to join the conversation?

- there is actually an easier way to factor ax^2+bx+c. All you have to do is first write (x-blank)(x-blank) and then use your calculator and press MODE 5 3 and there you'll write the coefficients of the equation. Then they'll give you two values and these values are the blanks I mentioned above (dont forget to multiply the negative). Hope this helped:) (In equations where x^2 has a coefficient then you solve normally but if one of the values that the calculator gives you is a fraction then you just have to take the denominator as the coefficient of x)(57 votes)
- yes in igcse we used to use calculator for these stuff(2 votes)

- This one is complicated(35 votes)
- bro explained it in the hardest way possible(25 votes)
- this took me like an hour, but at least i understand it now(15 votes)
- the education system...(0 votes)

- This lesson can simplified in one step :

Mode/5/3(14 votes) - As a non-native English speaker I was glad to hear that Joe can't pronounce "similarly" either. I've always had trouble with.(12 votes)
- me over here like “what are quadratics” a day before my test bc I had to self teach myself math the last four years and somehow completely missed this whole topic…(11 votes)
- I think it's disappointing that you do not even explain how we acquire this method of factorizing quadratics. Where is its derivation? Without understanding that one can only learn how to plug values, any question that tests comprehension will not be easy to solve.(9 votes)
- Hello asians:)(8 votes)
- Had to watch the organic chemistry tutor instead for this one 😭(6 votes)