Tim 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.
Create a correct multiplication table.
Compare the correct table with the given table to identify the incorrect entry.
The expression 6 x should be 6 x y .
Explanation
Analyzing the Problem Let's analyze the multiplication of the binomial ( 3 x + y ) by the trinomial ( x 2 + 2 y + 4 ) . We need to check each entry in the given multiplication table to identify the error.
Performing the Multiplication First, let's multiply 3 x 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 Now, let's multiply y by each term of the trinomial:
y × x 2 = x 2 y
y × 2 y = 2 y 2
$y \times 4 = 4y
Identifying the Error Now, let's create the correct multiplication table:
x 2
2 y
4
3 x
3 x 3
6 x y
12 x
y
x 2 y
2 y 2
4 y
Comparing this with the given table:
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 expression 6 x in the given table should be 6 x y .
Correcting the Chart Therefore, the change that can be made to correct the chart is: The expression 6 x should be 6 x y .
Examples
Understanding binomial and trinomial multiplication is crucial in various fields, such as physics and engineering, where complex equations often need simplification. For instance, when calculating the volume of a rectangular prism with sides defined by binomial expressions, you need to multiply these expressions. Correctly multiplying these expressions ensures accurate calculations, which can be vital in designing structures or predicting physical behaviors. This skill also forms the basis for more advanced algebraic manipulations and problem-solving techniques.
