Factoring + bx + c

To factor this trinomial, you need to find two integers that have a product equal to c and a sum equal to b. If there are no two integers that work, then the polynomial cannot be factored.

Examples

+ 11x +10

Step 1: Find two numbers that multiply to get 10 and add to get 11.

Step 2: Once you have found the two numbers that work, write out the polynomial like this:

(x +10)(x + 1)

Step 3: This shows the factored version of the trinomial + 11x +10

+ 6x + 4

Step 1: Find the GCF that all three numbers have in common.

Step 2: Create a box visual to help show your work in factoring the polynomial.

Step 3: Write out the factored polynomial and find two numbers that have a have a product equal to 2 and a sum equal to 3.

2(x+2)(x+1)

Why?

To know how to factor polynomials by inspection is a great skill to have because it can help solve math problems that include polynomials in the future and will make the process easier and quicker.

Vocabulary

Polynomial- A polynomial is an expression that is composed of variables, constants, and exponents, that are combined using mathematical operations including: addition, subtraction, multiplication and division.

Trinomial- A polynomial consisting of three terms.

GCF- Greatest common factor

Multiplying two binomials: distributive property

There are many different quick and efficient ways to multiply two binomials. One of them being distributive property.

Examples

(x-6)(x+12)

Step 1: Multiply the x in the first bracket with the x in the second bracket.

Step 2: Multiply the x with the 12.

Step 3: Multiply the -6 with the x in the second bracket.

Step 4: Multiply the -6 with 12.

Step 5: Write the products of the numbers in a line. This is the expanded form.

Step 6: Simplify. Add the like numbers together.

Multiplying two Binomials: Area Model

Using an area model to multiply two binomials is an easy and a great visual way to do so.

(2x+3)(x+12)

Step 1: Create a rectangular diagram and write out the monomials, one on top and one on the side.

Step 2: In the first box you put the product of x multiplied by 2x.

Step 3: In the second box you put the product of x multiplied by 3.

Step 4: Do the same with the bottom two boxes. The bottom left box is the product of 12 and 2x, the right box is the product of 12 and 3.

Step 5: Write the numbers in the boxes in a line. This is expanded form.

Step 6: Simplify.

a binomial is a polynomial that is the sum of two terms,

Why?

It is important to know the different ways in making multiplying binomials easy and quick. It will help in polynomial related math problems in the future and make it much easier for you to do the steps in solving the problems.

Vocabulary

Binomial- A polynomial that consists of two terms.

Trigonometry

The study of triangles and the relationships between sides and angles in a triangle.

Names of sides

Hypotenuse- The hypotenuse of a right triangle is always the side opposite of the right angle. It is the longest side in a right triangle.

Opposite- The opposite side is the side across from a given angle.

(The given angle is c)

Adjacent- The adjacent side is the non-hypotenuse side that is next to a given angle.

Why?

Knowing the names of a right triangle is important because it is crucial in determining the ratios of a triangle. It will also help when you’re using pythagorean theorem to solve math problems, if you know the sides of a right triangle it will be easier to identify the hypotenuse and make the process quicker in solving trigonometry equations.

Vocabulary

Theta- A variable to represent a measured angle

Right triangle- A triangle with one right angle, that is 90°

Exponent Laws

Laws that are used for simplifying expressions with exponents. These laws are helpful in simplifying expressions that have decimals, fractions, irrational numbers, and negative integers as their exponents. It helps makes the process much easier.

Multiplication Law– When multiplying like bases, keep the base the same and add the exponents.

Division Law– When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.

Power Law– To raise a number with an exponent to a power, you multiply the exponent and the power.

Zero Exponent Law- Any number raised to the power of 0 is equal to 1.

Why?

Exponent laws are important to know so that you can easily solve many mathematical problems involving repeated multiplication. The exponent laws simplifies the divisions and multiplication operations to help solve the problems and simplifies the calculations.

Vocabulary

Exponents– A short way to write repeated multiplication, written in exponential form such as

Powers– A power is a number written in exponential form. Consisting of a base and an exponent.

Prime Factorization 

A prime factorization of a composite number is a way of expressing the composite number as a product of its prime factors. We can use a tree diagram to find the prime numbers of the composite number we are trying to factorize.

Examples

Tree Diagram and Division Table

  or   

  or   

Tree Diagram Steps

Step 1: Write 60 on the top of the tree.

Step 2: Choose a pair of factors for 60 as branches.

Step 3: Choose a pair of factors for 6 and 10.

6 = 3 x 2

10 = 5 x 2

Step 4: 2, 2, 3, and 5 are on the last row and are the prime factors.

Division Table Steps

Step 1: Divide 24 by the smallest prime number, the smallest prime number should divide the number exactly, which is 2 in this case

Step 2: Divide the quotient, which is 12, by the smallest prime number

Step 3: Repeat the process until the quotient becomes 1

Step 4: Lastly, once you have all the prime factors, multiply them together.

2 x 2 x 2 x 3 = 24

Why?

It is important to know prime factorization because it can help you find the least common multiple (LCM). Which is an important skill to know in math and will make solving math problems much easier and efficient for you. We also use prime factorization on a daily basis which can be used in money exchange, calculating costs, time estimation, etc. I used a tree diagram as an example because it is an easy way to find prime factors of composite numbers and a great way if you are a visual learner.

Vocabulary

Factors– A number that divides another number

Factor pairs– A set of two whole numbers and when multiplied it results in a specific product

Prime number– A whole number which has exactly two factors, and the two factors are always 1 and the number itself

Composite number– A whole number that has more than two factors

Prime factors– Prime factors of a whole number are the factors of the number which are prime

Division Table photo source:

Admin. “Prime Factorization – Definition, Methods, Examples, Prime Factorize.” BYJUS, BYJU’S, 30 Nov. 2021, https://byjus.com/maths/prime-factorization/.