MATH CLASS: EXPONENTS AND POLYNOMIALS (TUTOR DANIEL ONLY)

Objectives:

· Identify and use the rules for exponents

· Apply operations with polynomials

· Use the FOIL method

· Use and apply scientific notation

Exponents

32

Here the 3 is called the base and the 2 is the exponent. The 2 represents the number of times the 3 will be multiplied. So 32= 3*3 or 9.

In another example, 43, it is read as ‘4 to the power of 3’ or ‘4 cubed’. It is represented mathematically as 4*4*4= 64.

The product property of exponents

If you multiply expressions with the same base, keep the base and add the exponents.

x2 * x4 = x6 34 * 35 = 39

The quotient property of exponents

If you divide expressions with the same base, keep the base and subtract the exponents.

x10 = x10-2 = x8 x3 = 1 = 1

x2 x8 x8-3=5 x5

Power to a power

When a power is raised to a power, multiply the exponents together.

(x2)6 = x12 (2x)3 = 8x3

Zero exponent property

Anything to the 0 power is equal to 1.

60=1 x0=1 1,00010=1

Negative exponent property

When there is a negative exponent, reverse the location of the power in the fraction.

4-2 = x-5 = = 32

Polynomials

Polynomial: An expression that consists of one or more terms, including some variable(s).

For example: 5x2 + 3x + 1.

Degree of a polynomial: The highest degree of any term in the polynomial.

For example: 5x2 + 3x + 1…the degree is 2

4x5-2x6-x2…the degree is 6

ab+9….the degree is 2 (1 for the a and 1 for the b)

Monomial: A polynomial with one term (mono means “one” like monologue). The following are all monomials:

5x3, 8, 4xy

Binomial: A polynomial with two terms which are not like terms . (bi means “two” like bicycle) The following are all binomials:

2x – 3

3x5 + 8x4

2ab – 6a2b5

Trinomial: A polynomial with three terms which are not like terms (tri means “three” like tricycle). The following are all trinomials:

x2 + 2x - 3

3x5 - 8x4 + x3

ax2 + bx + c

Linear polynomial: A polynomial with degree 1. Linear means “line”. These graphs form a diagonal, vertical, or horizontal line.

For example, the following are all linear polynomials:

y=3x + 5 (slope-intercept form)

y = ½x

x=a (vertical line, slope undefined)

y=b (horizontal line, slope=0)

Quadratic polynomial: A polynomial with a degree of 2 (2 is the biggest exponent). These graphs form parabolas or U’s (in chapter 7).

x2 + 8x – 5

a2 + 7a

ax2+ bx+c (quadratic formula)

Add or Subtract Polynomials

Add or subtract the like terms, paying close attention to the signs. Unlike terms cannot be combined.

Examples:

Adding

(2x3 + 3x2 – 4x + 7) + (x3 – 4x2 + x – 2) 2x3 + 3x2 – 4x + 7 + x3 – 4x2 + x – 2 Combine like terms. They are highlighted here.

3x3 - 1x2 – 3x + 5

Subtracting

(4x3 – 3x2 + 9x) – (2x3 – 8x + 12)

A negative is the same as -1. Distribute to all terms to the right of the negative.

(4x3 – 3x2 + 9x) – 1(2x3 – 8x + 12)

Change all the signs in the second parentheses and combine like terms.

(4x3 – 3x2 + 9x) + (-2x3 + 8x - 12)

(4x3 – 3x2 + 9x) + (-2x3 + 8x - 12)

 2x3 – 3x2 + 17x – 12

Multiply Polynomials

Multiply each term from the first parentheses or term to every term in the other polynomial. Combine like terms in the last step. Also see the FOIL example below.

Example:

–3x (2x2 – x + 7)

Distribute the term outside of the parentheses to everything within it.

–3x (2x2) – 3x (–x) – 3x (7)

–6x3 + 3x2 – 21x

Divide Polynomials

Divide each term in the polynomial by the denominator.

6x3 – 12x2 + 18x

2x

Divide all terms in the numerator by 2x.

3x2 – 6x + 9

FOIL

FOIL means first, outer, inner, last. This is the order in which to multiply two binomials (parentheses with two terms each).

Example: (x+3) (x-4)

First: x * x= x2

Outer: x * -4= -4x

Inner: 3 * x= 3x

Last: 3 * -4= -12

x2 -4x+3x-12 Combine like terms

x2-1x-12 You don’t have to write the 1 in front of the x. It is understood.

x2-x-12

**Review the background info page for more examples of FOIL.**

Scientific Notation When we use very small or large numbers, we need a better way to write them. Scientific notation is the way. Here is an example:

1,265,890

1. Convert the number to a decimal between the numbers 1 and 9.9.

Answer: 1.265890

2. Count the number of spaces you moved the decimal to the left (remember that the decimal is behind the 0).

Answer: 6

3. This number is your exponent in your base of 10.

Answer: 1.26589 x 106 (Note: You don’t need the last 0 if it is at the end.)

· Check it

1.265890 x 106

1.265890 x 1,000,000

= 1,265,890

It works for decimals too. The difference is that the exponent will be negative (-) since you have to move in the opposite direction.

Example:

0.0072134

1. Convert the number to a decimal between the numbers 1 and 9.9.

Answer: 7.2134

2. Count the number of spaces you moved the decimal to the right (now the decimal is in front of the number).

Answer: 3

3. This number is your exponent in your base of 10.

Answer: 7.2134 x 10-3

· Check it

7.2134 x 10-3

7.2134 x 0.001

= 0.0072134