Kate multiplied two binomials using the distributive property. She made a mistake in one of the steps. Where did she first make a mistake?
[tex](x-2)(3 x+4)[/tex]
1. [tex](x-2)(3 x)+(x-2)(4)[/tex]
2. [tex](x)(3 x)+(-2)(3 x)+(x)(4)+(-2)(4)[/tex]
3. [tex]3 x^2-6 x+4 x-8[/tex]
4. [tex]3 x^2-2 x-8[/tex]
A. Step 4
B. Step 3
C. Step 2
D. Step 1
Answer (2)
The mistake occurred in Step 1, where Kate incorrectly distributed the term (3x+4) as (x-2)(3x) + (x+2)(4) instead of (x-2)(3x) + (x-2)(4). Therefore, the correct answer is D.
Explanation
Problem Analysis Let's analyze Kate's steps to find the mistake in multiplying the binomials ( x − 2 ) ( 3 x + 4 ) . We need to carefully check each step to see where the error occurred.
Identifying the Error in Step 1 Step 1 claims: ( x − 2 ) ( 3 x + 4 ) = ( x − 2 ) ( 3 x ) + ( x + 2 ) ( 4 ) .
Let's correctly apply the distributive property: ( x − 2 ) ( 3 x + 4 ) = ( x − 2 ) ( 3 x ) + ( x − 2 ) ( 4 ) .
Comparing this with Kate's step 1, we see that she wrote ( x + 2 ) ( 4 ) instead of ( x − 2 ) ( 4 ) . This is where the mistake occurred.
Conclusion Since we've identified the mistake in Step 1, we don't need to analyze the subsequent steps.
Examples
Understanding binomial multiplication is essential in various fields, such as physics and engineering, where complex equations often need simplification. For instance, when calculating the area of a rectangular garden with sides (x - 2) and (3x + 4), you multiply these binomials. Correctly expanding this product ensures accurate area calculation, preventing errors in resource allocation and garden design. This skill also forms the basis for more advanced algebraic manipulations and problem-solving.
Kate made her first mistake in Step 1, where she incorrectly distributed the binomials. Instead of writing ( x − 2 ) ( 4 ) , she mistakenly wrote ( x + 2 ) ( 4 ) . Therefore, the correct answer is D.
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