Polynomials
Exponents
Zero and Negative Exponents
If a is a non-zero real number, then
a 0=1 ; a − 1 = 1aand
Exponents
Exponents
-
Exponents
Exponents
Exponents
Exponents
Exponents
=243x12
Exponents
a5
a2 = aaaaa aa
3= a = a 5− 2
Exponents
Exponents
Exponents
= 64
Exponents
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Exponents
ExponentsTry these:
Exponents
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