MATH1000 Section1.3

1. Math 1000
Stuart Jones
Section 1.3 Algebraic Expressions

2. Math 1000
Stuart Jones
In this section, we will begin our work with polynomials. A
polynomial is a sum or difference of monomials. A monomial is
a variable raised to some exponent, possibly multiplied by a
number, called a coefficient. Examples of polynomials are:
2x3 − 4x + 7
6
5x5 − 6x3 − 2x
−5x4
Note: Polynomials only contain integer exponents and do not
have any variables in the denominator.

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We can add, subtract, multiply, and divide polynomials. In this
section, we will look at the first three – we will come back to
division later. Adding and subtracting polynomials involves
getting like terms.

4. Math 1000
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Evaluate (6x2 − 4x + 2) + (−3x2 − 4x − 5)

5. Math 1000
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Evaluate (6x2 − 4x + 2) + (−3x2 − 4x − 5)
Answer:
3x2
− 8x − 3

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Evaluate (−9x5 − 3x3 − 1) − (4x5 + 2x2 − x − 4)

7. Math 1000
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Evaluate t4(t2 − 4t − 2) − t(t4 − 1)

8. Math 1000
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Evaluate y2(y + 5) + y2(y − 2)

9. Math 1000
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To multiply polynomials, we use the Distributive Property.
When we have binomials (2 terms), we frequently refer to it as
FOIL.

10. Math 1000
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Evaluate (x − 6)(x + 4)

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Evaluate (2x + 7)(x − 9)

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Evaluate (x − 5)(2x2 + 2x − 4)

13. Math 1000
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We often want to reverse FOIL, and this is called ”factoring”.
The most common factoring we will do is of trinomials (3
terms).
ax2
+ bx + c
Here is the basic process:
Multiply a times c. Find all factors of this number, listing
them as pairs, ignoring negative signs.
Find such a pair that can add OR subtract in some
combination to give you b.
Rewrite ax2 + bx + c so that the bx is now using the pair
of numbers from the previous step, instead of b.
Group together the first 2 terms, then the second two
terms. Pull out common factors in both sets.
The parentheses on both should be the same. Factor it
out to create your two factored binomials, your answer.

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Factor x2 − 2x − 15.

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Factor x2 − 2x − 15.
Answer: We find factors of ac (1 times 15), which is 15.
These are 1, 15 and 3,5. We need such a pair to add/subtract
to give us -2 (the b term). The only combination that works is
3 and -5. So we proceed:
x2
+ 3x − 5x − 15
(x2
+ 3x) + (−5x − 15)
x(x + 3) − 5(x + 3)
(x + 3)(x − 5)

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Facxtor x2 − 12x + 35

17. Math 1000
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Factor 15x2 − 57x + 54

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Factor 15x2 − 71x + 70

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There are some special factoring situations we should know.
Have these memorized!
Special Factoring Formulas
1 a2 − b2 = (a + b)(a − b)
2 a2 + 2ab + b2 = (a + b)2
3 a2 − 2ab + b2 = (a − b)2

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Factor 9 − y2

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Factor x2 − 4x + 4

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Factor 4x2 + 24x + 36

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The Bottom Line
Factoring is SUCH an important concept in this class and
EVERY other math class you’ll see in the future. You
MUST learn how to do this without struggling at all; it’s
extremely important.
Special factoring formulas? Memorize!