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Polynomials

Exponents

Zero and Negative Exponents

If a is a non-zero real number, then

a 0=1 ; a − 1 = 1aand

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-

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=243x12

Exponents

a5

a2 = aaaaa aa

3= a = a 5− 2

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= 64

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ExponentsTry these:

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3

PolynomialsA term is a constant, variable or a product or quotient of constants and variables. In an algebraic expression, terms are separated by + or – operators.

A monomial is an integer, variable or a product of integers and variables with non-negative exponents. Monomials are a subset of terms.Examples: 8, w, 24x3y

A polynomial is a monomial or sum of monomials. Examples: 5w + 8, -3x2 + 2x + 4, x, 0, 75y6

PolynomialsExample:Identify the terms of the polynomial:

7p4 – 3p3 + 5.

7p4 – 3p3 + 5 = 7p4 + -3p3 + 5

The terms are: 7p4, -3p3, 5

PolynomialsA polynomial that is composed of two terms is called a binomial.Examples: 3x + 15; 3x2 + 15xy; -15x - 6

A polynomial with exactly three terms is a trinomial.Examples; 3x2 + 15x -6; 2x + 3y + 15

We call polynomials with more than three terms polynomials. They have no special name.

PolynomialsThe part of a term that is a constant factor is the coefficient of that term.

Example: Identify the coefficient of each term in the polynomial. 5x3 - 6x2 + x – 11

Solution:The coefficient of 5x3 is 5. The coefficient of -6x2 is -6.The coefficient of x is 1, since x = 1x. The coefficient of -11 is simply -11.

Polynomials

PolynomialsThe leading term of a polynomial is the term with the highest exponent (degree). This will be the first term of the polynomial in standard form.

Its coefficient is called the leading coefficient. The degree of the leading term is the degree of the polynomial.

Example: 5x2 – 9x3 + 3x4 + 8x -15

Standard Form: 3x4 – 9x3 + 5x2 + 8x – 15 Leading Term: 3x4

Leading Coefficient: 3

Degree of the Polynomial: 4

Operations on PolynomialsTerms which have the same variable and powers are said to be like terms.For instance: 3x2y and 5x2y are like terms.

The powers must be the same for each variable. For instance: 3x2y and 3xy2 are not like terms.

Since the variable part of each like term represents the same number, you can collect (add and subtract) like terms.

Example: Simplify 3x3 + 2x + 5x3

3x3 + 2x + 5x3 = 8x3 + 2x

Operations on PolynomialsExamplesCombine like terms:

a) 5y3 – 9y4

b) 5x5 + 9 + 3x3 + 6x3 – 13 –6x 5

Operations on PolynomialsExamplesCombine like terms:

a) 5y3 – 9y3 = -4y3

b) 5x5 + 9 + 3x3 + 6x3 – 13 –6x 5

= -x5 + 9x3 - 4

Operations on PolynomialsWe can evaluate an expression’s numeric value for particular values of the variable(s)

Operations on PolynomialsWe add polynomials by gathering like terms:

= -6x3 + 7x – 2 + 5x3 + 4x2 + 3

= -x3 + 4x2 + 7x +1

Operations on Polynomials

Operations on Polynomials

= x5 – 3x2 + 5 + 5x5 – 2x4 + x2 – 15

= 6x5 – 2x4 -2x2 - 10

Operations on Polynomials

Operations on PolynomialsThe Opposite (Additive Inverse) of a Polynomial

The opposite of 6 is -6 because 6 + -6 = 0.

The opposite of 6x is – 6x because 6x + -6x = 0.

The opposite of 6x - 3 is -6x + 3, because (6x - 3) + (-6x + 3) = 0.

Note that we can find the opposite of a polynomial by changing the sign of each term.

Operations on PolynomialsThe Opposite (Additive Inverse) of a Polynomial

Example: Find the opposite of 3x2 – 2x + 1

→ -3x 2 + 2x – 1

We can also write this as-1(3x2 – 2x + 1) or -(3x2 – 2x + 1).(We are actually getting a little ahead of ourselves here since we have not talked about distribution and polynomials yet, but we’ll rely on our assumed knowledge of distribution in arithmetic for understanding.)

Operations on PolynomialsSubtraction of Polynomials

We can subtract one polynomial from another by adding the opposite of the polynomial being subtracted.

Subtract (9x4 +2x3 -10x) – (5x4 – 4x3 + 4)

= 9x4 +2x3 -10x – 5x4 + 4x3 –4

= 4x4 + 6x3 – 10x - 4

Operations on PolynomialsSubtraction of Polynomials

Try this:Subtract: (10x5 – 2x3 + 5x2) - (-4x5 + 2x3 – 7x2)

Operations on PolynomialsSubtraction of Polynomials

Try this:Subtract: (10x5 – 2x3 + 5x2) - (-4x5 + 2x3 – 7x2)

= 10x5 – 2x3 + 5x2 + 4x5 - 2x3 + 7x2

= 14x5 – 4x3 + 12x2

Operations on PolynomialsSubtraction of Polynomials

Operations on Polynomials

Operations on Polynomials

(6)(7)(x)(x) = 42x2

(5)(-1)(a)(a) = -5a 2

(-8)(3)(x6)(x4) = -24x10

Operations on PolynomialsMultiplying Polynomials: polynomial by a monomial

Multiply 6x2(x3 – 3x2 – 6x + 4)

Use distribution: 6x2(x3 – 3x2 – 6x + 4)

= 6x2(x3) – 6x2(3x2) – 6x2(6x) + 6x2(4)

= 6x5 – 18x4 – 36x3 + 24x2

Operations on Polynomials

Operations on Polynomials

= 3x2(5x5) – 3x2(2x4) + 3x2(x2) – 3x2(15)

= 15x7 – 6x6 + 3x4 - 45x2

Operations on PolynomialsMultiplying Polynomials: Two Binomials

To multiply two binomials, distribute each term in the first binomial over the second binomial.

= 3x(2x) + 3x(-1) +2(2x) + 2(-1)= 6x2 - 3x + 4x -2= 6x2 + x – 2

This is often called the FOIL method because youmultiply the First, Outer, Innerand Last terms.F O

I L

Operations on PolynomialsMultiplying Polynomials: Two Binomials

Try these:

(x + 5)(x + 7) =

(x - 5)(x + 7) =

(x + 5)(x – 7) =

(x – 5)(x – 7) =

Operations on PolynomialsMultiplying Polynomials: Two Binomials

Try these:

(x + 5)(x + 7) = x(x) + (x)7 + 5(x) + 5(7) = x 2 + 12x + 35

(x - 5)(x + 7) = x(x) + (x)7 - 5(x) - 5(7) = x 2 + 2x - 35

(x + 5)(x – 7) = x(x) + (x)(-7) + 5(x) + 5(-7) = x 2 - 2x - 35

(x – 5)(x – 7) = x(x) + (x)(-7) - 5(x) – 5(-7) = x 2 - 12x + 35

Operations on PolynomialsMultiplying Polynomials: Two Binomials

One model you may encounter for the multiplication of two binomials is the Area of a Rectangle:

Operations on PolynomialsSpecial Products:

There are some patterns of products that occur often enough that it is useful to memorize a solution pattern for them. You would still get the same answer if you foiled, but knowing the pattern can save time:

Difference of two squares: (a + b)(a – b) = a 2 – b2

Square of a binomial sum: (a + b) 2 = a2 + 2ab + b 2

Square of a binomial difference: (a - b)2 = a2 - 2ab + b2

Special Products:

Difference of two squares: (a + b)(a – b) = a2 – b2

Square of a binomial sum: (a + b)2 = a2 + 2ab + b2

Square of a binomial difference: (a - b)2 = a2 - 2ab + b2

Try these:

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

(2x + 5)2 =

(2x - 5)2 =

Operations on Polynomials

Operations on PolynomialsSpecial Products:

Difference of two squares: Square of a binomial sum: Square of a binomial difference:

(a + b)(a – b) = a 2 – b2

(a + b) 2 = a2 + 2ab + b 2

(a - b)2 = a2 - 2ab + b2

Try these:

(2x + 5)(2x - 5) = (2x)2 - 52 = 4x2 – 25

(2x + 5)2 = (2x)2 + 2(2x)(5) + 52 = 4x2 + 20x + 25

(2x - 5)2 = (2x)2 - 2(2x)(5) + 52 = 4x2 - 20x + 25

=

Operations on PolynomialsDividing a polynomial by a monomial

To divide a polynomial by a monomial, divide each term by the monomial.

x5+12 x3+18 x2

6 xx5 x3 x2

=6 x

+126 x

+186 x

x42

6+2 x +3 x

=

Operations on PolynomialsDividing a polynomial by a monomial

Sometimes when we divide a polynomial by a monomial, we get a result that is not a polynomial, a rational function. We will study these later.

x5+12 x3+18 x2

6 x5

x5 x3 x2

= +12 +186 x5 6 x5 6 x5

1 2 36

+ x2 + x3