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Polynomials
Polynomials• A polynomial is a term or the sum or
difference of two or more terms.• A polynomial has no variables in the
denominator.• The “degree of a term” is the exponent of
the variable (4x3 is a 3rd degree term).• The “degree of the polynomial” is the
same as the degree of the term with the highest degree.
(x5 + 4x3 – 3x + 2 is a fifth degree polynomial)
• Polynomials in standard form are in order of degree from highest to lowest with the constant at the end.
Polynomials By Terms
• A polynomial with one term is called a monomial.
• A polynomial with two terms is called a binomial.
• A polynomial with three terms is called a trinomial.
Polynomials by Degrees
• A first degree polynomial is linear.• A second degree polynomial is
quadratic.• A third degree polynomial is cubic.• A polynomial with no variable is
called a constant.
Examples 1 and 2• Name the degree of each term and each
polynomial. Put them in standard form.
• Degree of each term 5, 3, 1, and 0• Degree of the polynomial 5• It’s in standard form.• New Problem:
• Degree of each term 1, 2, 0, and 3• Degree of the polynomial 3rd
• In standard form, it is:
x x x5 34 3 2
5 2 3 92 3x x x
9 2 5 33 2x x x
Model Polynomial Addition and Subtraction
Algebra Tiles
1 unit or -1 unit
x units or-x units
x2 units or-x2 units
Model the following with Algebra Tiles• (2x2 – x) + (x2 + 3x – 1)
3x2 + 2x - 1
(2x2 + 6) – 4x2
-2x2 + 6
What is the expression modeled below?
(2x2 – 2x – 4) + (-x2 + 3x + 2)
Adding Polynomials
• Collect like terms. • In order to have like terms,
the variable parts must be exactly the same.
• Combine the coefficients (the numbers in front of the variable).
Subtracting Polynomials
• Drop the first set of parentheses.
• Distribute a –1 in the second set of parentheses.
• Combine like terms.
To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7
Subtract
• (3x2 + 2x – 1) – (x2 + 4x – 2)• Distribute the -1• 3x2 + 2x – 1 - x2 - 4x + 2• Combine like terms
2x2 – 2x + 1
An egg is thrown off the top of a building. Its height in meters above the ground can be approximated by the polynomial 300 + 2t – 4.9t2, where t is the time since it was thrown in seconds.
How high is the egg above the ground after 5 seconds?
How high is the egg above the ground after 6 seconds?
Example 3
187.5
135.6
Example 4
• A firework is launched from a platform 6 feet above the ground at a speed of 200 feet per second. The firework has a 5-second fuse. The height of the firework in feet is given by the polynomial -16t2 + 200t + 6, where t is the time in seconds. How high will the firework be when it explodes?
606
Try these…
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15
2. 25x2 – 3x4
Write each polynomial in standard form. Then
give the leading coefficient.
3. 24g3 + 10 + 7g5 – g2
4. 14 – x4 + 3x2
4
5
–x4 + 3x2 + 14; –1
7g5 + 24g3 – g2 + 10; 7
Try these…
Classify each polynomial according to its degree and number of terms.
5. 18x2 – 12x + 5 quadratic trinomial
6. 2x4 – 1 quartic binomial
7. The polynomial 3.675v + 0.096v2 is used to estimate the stopping distance in feet for a car whose speed is v miles per hour on flat dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? 727.65 ft
Try these…
• (5x2 – 2x + 3) + (6x2 + 5x + 6)
• (3x2 – x + 3) + (2x2 + 2x + 9)
• (11x2 + 3x – 1) – ( 2x2 + 2x + 8)
• (x2 – x + 2) – (-3x2 – x – 4)
11x2+3x+9
5x2+x+12
9x2+x-9
5x2+6