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### Polynomials

Lesson 3.3 Factoring

Polynomials

- A math equation consisting of one to many terms.
- Examples:
- 6, x, 6x, -1/2xy, 2y + x, x2 – 5x - 9
- Polynomials cannot have a variable as a denominator nor negative exponents.

- Are the following polynomials?
- 7/a
- ¼ xy – 10
- 3pq1/2
- √7 x4 – x3
- 8-2

- Polynomials with
- one term are called monomials
- 5x3, 8, x2, etc
- two terms are called binomials
- 3x – 1, 2x2 + 8, etc
- three terms are called trinomials
- 2x2 – 4x + 9

- Variables – a letter that represents one or more numbers
- 4y = y is the variable
- Coefficient – number in front of a variable
- 4y = coefficient is 4

Degrees of a polynomial

- The degree of a polynomial is the degree of the term with the highest exponent.
- Constant term: term without a variable.

- 2x – 1 = degree of 1 Constant term of -1 These are called a linear.
- 2x2 + 8 = degree of 2 Constant term of 8 These are called quadratic.
- 2x3 – 5 = degree of 3 Constant term of -5 These are called cubic.

Example 1 a

- State the degree, coefficient’s and constant term of the polynomial.
- 5x3 + x2 – 7x + 9

Example 2 a

- State the degree, coefficient and constant term of the polynomial.
- 6a – 4a2 - 3

Adding and Subtracting polynomials a

- Find like terms and combine them in order to simplify polynomials.
- 4x – 2x2 + 3 – 6x2 + 5 – x

Try the following a

- a2b – ab2 + 4a3b – 7ab2 + 5a2b
- (3a – 4b + c) + (3b – 5c – 3a)

Be Careful with subtraction a

- (4x2 – 9x + 6) – (2x2 – 3x – 1)

Factoring Linear polynomials a

- Just as natural numbers can be factored so can polynomials.
- Find the GCF in each term and then factor.

Factoring Examples a

- 4m + 12
- GCF = 4
- = 4 (m + 3)

- 6 – 15a a
- GCF = 3
- = 3 (2 – 5a)

Try the following a

- 6n + 9 =
- 6c + 4c2 =
- 3g + 6 =
- 8d + 12d2 =

Factoring Trinomials a

- ax2 + bx + c
- 5 – 10z – 5z2
- Find the GCF of all three terms.
- In this example the GCF is 5.
- Factor out a 5 from each and write as a product.
- 5 ( 1 – 2z – z2)

Examples a

- 18a2 – 12a + 6
- 9 + 27x – 45x2

Factoring with more than one variable a

- Find all GCF’s, numbers and letters.
- -12 x3y – 20xy2 – 16x2y2
- GCF for numbers = 4
- GCF for letters = 1x and 1y
- 4xy (-3x2 – 5y – 4xy)

- 5ab a 2 + 10a2b3 – 15a2b4

- - a 20c4d - 30c3d2 – 25cd

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