Multiplying integers can be modeled by repeat addition. The multiplication of integers can be represented on a number line.
Therefore, 3(–2) 6.
The number line below models the product 2 (–3).
What happens if the order of the factors is changed to (–2)3? The Commutative Property of
Multiplication guarantees that 3(–2) = (–2)3. Therefore, –2(3) = –6.
In 3(–2) 6 and –2(3) = –6, one factor is positive, one factor
is negative, and the product is negative. These examples suggest the following
rule for multiplying two integers with different signs.
Example
Find each product.
1.
6(–8)
Alternative Solutions :
6(–8) = –48 The
factors have different signs. The product is negative.
2.
–5(9)
Alternative Solutions :
–5(9)
= –45 The factors have different signs. The
product is negative.
You already know that the product of two positive numbers is
positive. What is the sign of the product of two negative numbers? Consider the
product –2(3).
0 = –2(0) Multiplicative Property of Zero
0 = –2[3
+ (3)] Replace 0
with 3 + (–3) or
any zero pair.
0 = –2(3) + (–2)( –3) Distributive
Property
0 = –6
+
? –2(3) = –6
By the Additive Inverse Property, –6 + 6 = 0. Therefore, –2(–3) must be equal to 6. This example
suggests the following rule for multiplying two integers with the same sign.
Example
Find each product.
3.
15(2)
Alternative Solutions :
15(2) =
30 The factors have the same sign.
The product is positive.
4.
–5(–6)
Alternative Solutions :
–5(–6) = 30 The factors have the same sign. The product is positive.
To find the product of three or more numbers, multiply the
first two numbers. Then multiply the result by the next number, until you come
to the end of the expression.
Example
Find each product.
5.
8(–10)( –4)
Alternative Solutions :
8(–10)(–4) = –80(–4) 8(–10) = –80
= 320 –80(–4) = 320
6.
5(–3)( –2)( –2)
Alternative Solutions :
5(–3)(–2)( –2) = –15(–2)( –2) 5(–3) = –15
= 30(–2) –15(–2) = 30
= –60 30(–2) = –60
You can use the rules for multiplying integers to evaluate
algebraic expressions and to simplify expressions.
Example
7.
Evaluate 2xy if x = –4 and y = –2.
Alternative Solutions :
2xy
= 2(–4)( –2) Replace
x with –4 and y with –2.
= –8(–2) 2(–4) = –8
= 16 –8(–2) = 16
8. Simplify (2a)(–5b).
Alternative Solutions :
(2a)(–5b) = (2)(a)(–5)(b) 2a = (2)(a); –5b = (–5)(b)
= (2)( –5)(a)(b) Commutative Property
= –10ab (2)(–5) = –10; (a)(b)
= ab
Example
Finance Link
9. The
graphs of A(3, 5), B(1, 2), and C(5, –1)
are connected with line segments to form a triangle. Multiply each x-coordinate by –1 and redraw the triangle. Describe how
the position of the triangle changed.
Alternative Solutions :
A(3, 5) → (3 × 1, 5) → A′ (–3, 5)
B(1, 2) → (1 ×1, 2) → B′ (–1, 2)
C(5, –1) → (5 × –1, –1) → C′ (–5, –1)
Triangle
A′B′C′ is shown in green.
It is
the same size and shape as triangle ABC,
but it is reflected, or flipped, over the y-axis.
Sumber
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