SOM.ORO MNTONEE SECOVSTIEN SCHERS
finishim
K. $\rightarrow$ ell que wans
i. Writo- the manks a parcs concer by
a) 2 xipminearn tipures
by 3 decimal phans
1. Factorix completeh
(a) $2 x+x y-y$
by $4-(a-1)^2$
A a Baliumalization Armenimane $\frac{2,3-2}{4+1}$
Answer (1)
Write 2.3 to 2 significant figures: 2.3
Write 2.3 to 3 decimal places: 2.300
Attempt to factor 2 x + x y − y : Expression remains as 2 x + x y − y
Factor 4 − ( a − 1 ) 2 using difference of squares: ( 3 − a ) ( 1 + a )
Rationalize 4 + 1 2.3 − 2 : 0.06
Explanation
Problem Overview We are given a problem with three parts: writing a number to specified precision, factoring two expressions, and rationalizing a simple expression.
Writing the Number First, we need to write the number 2.3 to 2 significant figures and 3 decimal places. Since 2.3 already has two significant figures, writing it to two significant figures remains 2.3. Writing it to three decimal places gives 2.300.
Factoring the First Expression Next, we factor the expression 2 x + x y − y . We can factor by grouping. However, there's no direct grouping possible to simplify the expression. Thus, the expression remains as 2 x + x y − y .
Factoring the Second Expression Now, we factor the expression 4 − ( a − 1 ) 2 . This is a difference of squares, which can be factored as A 2 − B 2 = ( A − B ) ( A + B ) . Here, A = 2 and B = ( a − 1 ) . So, we have:
4 − ( a − 1 ) 2 = [ 2 − ( a − 1 )] [ 2 + ( a − 1 )] = ( 2 − a + 1 ) ( 2 + a − 1 ) = ( 3 − a ) ( 1 + a )
Rationalizing the Expression Finally, we rationalize the expression 4 + 1 2.3 − 2 . First, simplify the numerator and the denominator:
4 + 1 2.3 − 2 = 5 0.3 = 0.06
Since there are no radicals, no further rationalization is needed. The simplified form is 0.06.
Final Results In summary:
2 significant figures: 2.3
3 decimal places: 2.300
2 x + x y − y remains as 2 x + x y − y
4 − ( a − 1 ) 2 = ( 3 − a ) ( 1 + a )
4 + 1 2.3 − 2 = 0.06
Examples
Factoring expressions is a fundamental skill in algebra and is used extensively in various fields such as engineering, physics, and computer science. For example, when designing a bridge, engineers use factoring to analyze the forces acting on the structure and ensure its stability. Similarly, in computer graphics, factoring is used to optimize the rendering of 3D models, making the process more efficient. Rationalizing expressions is useful in simplifying calculations in physics and engineering, especially when dealing with measurements and approximations.
