Flowchart Math

⤵

What is Factoring?

Factoring is when you rewrite an expression as multipliction.

For example, 8 = 4×2.

The factors are the numbers being multiplied.

Is there a GCF?

Look for a GCF. In other words, can ALL of the terms be divided by some number or by some combination of the same variables?

Find the GCF of all terms.

You want the biggest number and/or some combination of variables that will divide into every term in the expression.

Example: GCF

4x³ + 8x² - 20x

= 4x(x² + 2x - 5)

4x is the GCF so we can rewrite the original problem as 4x times whatever would multiply back to the original problem.

To check your answer, you can distribute to see if what you wrote equals the original problem. You should also make sure the expression in parentheses does not have anything else in common for all of its terms.

How many terms?

Count the number of terms in the expression.

The terms are the things being added/subtracted. Terms may look like 5x² or like 9x or just a number like 4 or just a variable like x.

7x³ + 9x² + x - 1 has a total of 4 terms.

Difference of Squares?

Difference of Squares

Difference of Squares is a subtraction problem involving two terms that are BOTH perfect squares.

Two examples would be x² - 49 and 9x² - 16.

Difference of Squares

You should see a subtraction problem involving two terms that are BOTH perfect squares.

Example

4x² - 25

= (2x + 5)(2x - 5)

The square root of 4x² is 2x, and the square root of 25 is 5. We can rewrite the problem as the product of (2x + 5) and (2x - 5). To check this result: distribute (some call this FOIL) the terms on the left to the terms on the right to see that it matches the original problem.

Sum or Difference of Cubes?

Look for either an addition problem or a subtraction problem involving two terms that are BOTH perfect CUBES.

Two examples would be x³ - 27 and 8x³ + 125y³.

Sum/Difference of Cubes

You should see EITHER an addition or a subtraction problem involving two terms that are BOTH perfect CUBES.

Sum of cubes formula: a³ + b³ = (a + b)(a² - ab + b²)

Difference of cubes formula: a³ - b³ = (a - b)(a² + ab + b²)

Example

x³ - 27

= (x - 3)(x² + 6x + 9)

The cube root of x³ is x, and the cube root of 27 is 3. Rewrite as the product of (x - 3) times the other part, according to the formula.

Check your answer using the distributive property.

Prime

It appears that this one cannot be factored.

Perfect Square Trinomial?

If the first and third terms are BOTH perfect squares, then this MIGHT be a perfect square trinomial.

For example, 4x² + 20x + 25.

Perfect Square Trinomial

The first and third term should be perfect squares.

The middle term should be twice the product of the two square roots.

Example

4x² + 12x + 9

= (2x + 3)²

Notice 12x is equal to twice 2x × 3.

Is b² - 4ac a perfect square?

Look at the 3 coefficients and find the value of b² - 4ac.

See if you get a perfect square (something you can take the square root of easily like 9 or 64).

Is a = 1?

The coefficient a is equal to 1 in the expression x² + 7x + 12

The coefficient a equals 3 in the expression 3x² + 4x + 9

Easy Trinomial

I call it an easy trinomial when a = 1.

Try to find 2 numbers that multiply to the c term and add up to the b coefficient.

Example

x² + 10x - 24

= (x + 12)(x - 2)

Notice 12 × -2 equals -24, and 12 plus -2 equals 10.

Hard Trinomial

I call it a hard trinomial when a ≠ 1.

List all pairs of numbers that multiply to equal c. Do the same for a. The trick is to find the right combination of c₁×a₁ plus c₂×a₂ to equal b.

Example

3x² + 14x + 8

8: 1×8, 2×4

3: 3×1

= (3x + 2)(x + 4)

Notice 4 × 3 plus 2 × 1 equals 14.

Use the distributive property (FOIL) to check your answer.

Prime

It appears that this one cannot be factored.

Factor by Grouping?

If the expression has 4 terms (or 6 or 8...), grouping might be possible.

Factor by Grouping

Try to put the expression into equal groups (usually groups of 2) and see if you can find a GCF for each group.

After taking out the GCF's of each group, look for matching expressions. If you see matching expressions, you likely can factor by grouping.

Example

x³ + 4x² + 8x + 32

put in groups (x³ + 4x²) + (8x + 32)

find GCF of each group x²(x + 4) + 8(x + 4)

Notice (x + 4) is a matching expression.

Write (x + 4) and the expression formed with the GCF's.

= (x + 4)(x² + 8)

Use the distributive property (FOIL) to check your answer.

Prime

Flowchart Math: Factoring

What is Factoring?

Factoring is when you rewrite an expression as multipliction.

For example, 8 = 4×2.

The factors are the numbers being multiplied.

Is there a GCF?

Look for a GCF. In other words, can ALL of the terms be divided by some number or by some combination of the same variables?

Find the GCF of all terms.

You want the biggest number and/or some combination of variables that will divide into every term in the expression.

Example: GCF

How many terms?

Count the number of terms in the expression.

The terms are the things being added/subtracted. Terms may look like 5x2 or like 9x or just a number like 4 or just a variable like x.

7x3 + 9x2 + x - 1 has a total of 4 terms.

Difference of Squares

Difference of Squares is a subtraction problem involving two terms that are BOTH perfect squares.

Two examples would be x2 - 49 and 9x2 - 16.

Difference of Squares

You should see a subtraction problem involving two terms that are BOTH perfect squares.

Example

Sum or Difference of Cubes?

Look for either an addition problem or a subtraction problem involving two terms that are BOTH perfect CUBES.

Two examples would be x3 - 27 and 8x3 + 125y3.

Sum/Difference of Cubes

You should see EITHER an addition or a subtraction problem involving two terms that are BOTH perfect CUBES.

Sum of cubes formula: a3 + b3 = (a + b)(a2 - ab + b2)

Difference of cubes formula: a3 - b3 = (a - b)(a2 + ab + b2)

Example

It appears that this one cannot be factored.

Perfect Square Trinomial?

If the first and third terms are BOTH perfect squares, then this MIGHT be a perfect square trinomial.

For example, 4x2 + 20x + 25.

Perfect Square Trinomial

The first and third term should be perfect squares.

The middle term should be twice the product of the two square roots.

Example

Is b² - 4ac a perfect square?

Look at the 3 coefficients and find the value of b2 - 4ac.

See if you get a perfect square (something you can take the square root of easily like 9 or 64).

Is a = 1?

The coefficient a is equal to 1 in the expression x2 + 7x + 12

The coefficient a equals 3 in the expression 3x2 + 4x + 9

Easy Trinomial

I call it an easy trinomial when a = 1.

Try to find 2 numbers that multiply to the c term and add up to the b coefficient.

Example

Hard Trinomial

I call it a hard trinomial when a ≠ 1.

List all pairs of numbers that multiply to equal c. Do the same for a. The trick is to find the right combination of c1×a1 plus c2×a2 to equal b.

Example

It appears that this one cannot be factored.

Factor by Grouping?

If the expression has 4 terms (or 6 or 8...), grouping might be possible.

Factor by Grouping

Try to put the expression into equal groups (usually groups of 2) and see if you can find a GCF for each group.

After taking out the GCF's of each group, look for matching expressions. If you see matching expressions, you likely can factor by grouping.

Example

It appears that this one cannot be factored.

The terms are the things being added/subtracted. Terms may look like 5x² or like 9x or just a number like 4 or just a variable like x.

7x³ + 9x² + x - 1 has a total of 4 terms.

Two examples would be x² - 49 and 9x² - 16.

Two examples would be x³ - 27 and 8x³ + 125y³.

Sum of cubes formula: a³ + b³ = (a + b)(a² - ab + b²)

Difference of cubes formula: a³ - b³ = (a - b)(a² + ab + b²)

For example, 4x² + 20x + 25.

Look at the 3 coefficients and find the value of b² - 4ac.

The coefficient a is equal to 1 in the expression x² + 7x + 12

The coefficient a equals 3 in the expression 3x² + 4x + 9

List all pairs of numbers that multiply to equal c. Do the same for a. The trick is to find the right combination of c₁×a₁ plus c₂×a₂ to equal b.