Chin made an error in his chart showing the multiplication of the binomial by the trinomial.
[tex]$(3 x+y)(x^2+2 y+4)$[/tex]
Which change can be made to correct the chart?
A. The expression [tex]$3 x^3$[/tex] should be [tex]$3 x^2$[/tex].
B. The expression [tex]$6 x$[/tex] should be [tex]$6 x y$[/tex].
C. The expression [tex]$x^2 y$[/tex] should be [tex]$x^2 y^2$[/tex].
D. The expression [tex]$4 y$[/tex] should be [tex]$4 y^2$[/tex].
Answer (1)
Multiply each term of the binomial by each term of the trinomial.
Compare the results with the entries in the chart to identify the incorrect entry.
Check each of the options to see which one corrects the chart.
The correct change is: The expression 6 x should be 6 x y .
Explanation
Understanding the Problem We are given a multiplication chart for the binomial ( 3 x + y ) and the trinomial ( x 2 + 2 y + 4 ) . We need to identify the error in the chart and determine which change will correct it.
Performing the Multiplication Let's multiply each term of the binomial by each term of the trinomial:
3 x ∗ x 2 = 3 x 3 3 x ∗ 2 y = 6 x y 3 x ∗ 4 = 12 x y ∗ x 2 = x 2 y y ∗ 2 y = 2 y 2 y ∗ 4 = 4 y
Identifying the Error Now, let's compare our results with the entries in the chart:
x 2
2 y
4
3 x
3 x 3
6 x
12 x
y
x 2 y
2 y 2
4 y
We can see that the entry corresponding to 3 x ∗ 2 y is incorrect. The chart shows 6 x , but the correct result is 6 x y .
Checking the Options Now, let's check the given options to see which one corrects the chart:
The expression 3 x 3 should be 3 x 2 . - Incorrect, 3 x ∗ x 2 = 3 x 3
The expression 6 x should be 6 x y . - Correct, 3 x ∗ 2 y = 6 x y
The expression x 2 y should be x 2 y 2 . - Incorrect, y ∗ x 2 = x 2 y
The expression 4 y should be 4 y 2 . - Incorrect, y ∗ 4 = 4 y
Conclusion Therefore, the correct change is: The expression 6 x should be 6 x y .
Examples
Understanding how to multiply polynomials and identify errors in multiplication charts is crucial in various fields, such as engineering and computer science. For example, when designing circuits or writing code, engineers and programmers often need to manipulate polynomial expressions to model system behavior or optimize performance. Being able to accurately expand and simplify these expressions, as well as quickly spot and correct mistakes, can save time and prevent costly errors in these applications. This skill also forms a foundation for more advanced mathematical concepts, such as calculus and differential equations, which are essential tools in many scientific and technical disciplines.
