Algebra 2 by Richard Wright
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Objectives:
SDA NAD Content Standards (2018): AII.4.2, AII.5.1, AII.6.4
A polynomial is an expression in the form anxn + an−1xn−1 + ⋯ + a2x2 + a1x + a0. Each group of anxn is called a term. The number multiplied by the x's, the an, is called a coefficient. The highest exponent of all the terms is called the degree and it gives important information about the shape of the graph as seen in lesson 3-04.
Adding and subtracting polynomials is often called “combining like terms.” Add or subtract the coefficients of terms with the same combination of variables and exponents.
To add or subtract polynomial expressions,
Simplify (x3 + 2x2 − 3x + 15) + (4x3 − 3x2 + x − 2).
Solution
Find the terms with the same combination of variables and exponents.
(x3 + 4x3) + (2x2 + (−3x2)) + (−3x + x) + (15 − 2)
Add the coefficients of the like terms.
5x3 − x2 − 2x + 13
Simplify (4x2 + 3x − 7) − (x3 + 4x − 8).
Solution
Find the terms with the same combination of variables and exponents.
(−x3) + (4x2) + (3x − 4x) + (−7 − (−8))
Add the coefficients of the like terms.
−x3 + 4x2 − x + 1
Multiplying polynomials uses the distributive property. That means multiplying every term of the first group with every term of the second group.
To multiply polynomial expressions,
Simplify 2x(x + 7).
Solution
Use the distributive property to multiply the 2x with both terms in the (x + 7).
2x(x + 7)
2x(x) + 2x(7)
2x2 + 14x
Simplify (x + 5)(x − 4).
Solution
Use the distributive property to multiply both the x and the 5 from the (x + 5) with both the x and the −4 from the (x − 4).
(x + 5)(x − 4)
x·x + x(−4) + 5x + 5(−4)
x2 − 4x + 5x − 20
Combine like terms.
x2 + x − 20
Simplify (2x + 1)(x2 − 3x + 4).
Solution
Use the distributive property to multiply both the 2x and the 1 from the (2x + 1) with each term in the other group.
(2x + 1)(x2 − 3x + 4)
2x(x2) + 2x(−3x) + 2x(4) + 1(x2) + 1(−3x) + 1(4)
2x3 − 6x2 + 8x + x2 − 3x + 4
Combine like terms.
2x3 − 5x2 + 5x + 4
Simplify (x − 3)2.
Solution
This is the same thing as (x − 3)(x − 3).
Use the distributive property to multiply both the x and the −3 from the first (x − 3) with each term in the other (x – 3).
(x − 3)(x − 3)
x(x) + x(−3) − 3(x) − 3(−3)
x2 − 3x − 3x + 9
Combine like terms.
x2 − 6x + 9
Simplify (x − 1)(x + 4)(x − 2).
Solution
Multiply the first two (or last two) groups first. Then multiply that result with the final group.
Use the distributive property to multiply both the x and the −1 from the (x − 1) with each term in the (x + 4).
(x − 1)(x + 4)(x − 2)
(x(x) + x(4) − 1(x) − 1(4))(x − 2)
(x2 + 4x − x − 4)(x − 2)
Collect like terms.
(x2 + 3x − 4)(x − 2)
Use the distributive property to multiply each term from the first group with both terms of the last group.
x2(x) + x2(−2) + 3x(x) + 3x(−2) − 4(x) − 4(−2)
x3 − 2x2 + 3x2 − 6x − 4x + 8
Collect like terms.
x3 + x2 − 10x + 8
166 #1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 29, 33, 37, Mixed Review = 20
Mixed Review
