Mark multiplied the expression 3x2−276x+30⋅x2+9x+20x2−x−12
incorrectly.
First, Mark factored each of the polynomial expressions: 3(x+3)(x−3)6(x+5)⋅(x+5)(x+4)(x+3)(x−4)
.
Next, Mark divided out common factors and got the answer: x+32
.
Determine Mark’s error.
Option #1: Mark factored incorrectly.
Option #2: Mark divided out common factors incorrectly.
Option #3: Mark did not fully simplify the expression.
Option #1: Mark factored incorrectly.
Mark factored the expression correctly. The error occurred in the division step where he obtained the incorrect answer of x+32.
Option #1: Mark factored incorrectly.
Based on the information provided, Mark's error can be determined as follows:
Option #1: Mark factored incorrectly.
To check if Mark factored incorrectly, we need to expand the factored expressions and see if they simplify back to the original expression. Let's expand the factored expressions:
3(x+3)(x−3) = 3x^2 - 9x + 3x - 9 = 3x^2 - 6x - 9
6(x+5)(x+5) = 6x^2 + 30x + 30x + 150 = 6x^2 + 60x + 150
(x+4)(x+3)(x−4) = (x^2 + 3x + 4x + 12)(x - 4) = (x^2 + 7x + 12)(x - 4) = x^3 - 4x^2 + 7x^2 - 28x + 12x - 48
The expanded expressions do not match the original expression, so Mark's error is not in the factoring of the terms.
Option #2: Mark divided out common factors incorrectly.
To check if Mark divided out common factors incorrectly, we can expand the factored expressions and see if we can simplify it further. Let's expand them:
3(x+3)(x−3) = 3x^2 - 9x + 3x - 9 = 3x^2 - 6x - 9
6(x+5)(x+5) = 6x^2 + 30x + 30x + 150 = 6x^2 + 60x + 150
(x+4)(x+3)(x−4) = (x^2 + 7x + 12)(x - 4) = x^3 - 4x^2 + 7x^2 - 28x + 12x - 48
When we simplify further, we get:
3x^2 - 6x - 9 + 6x^2 + 60x + 150 + x^3 - 4x^2 + 7x^2 - 28x + 12x - 48
= x^3 + 3x^2 + 29x + 93
This expanded expression is not the same as the answer Mark obtained (x+32), so Mark's error is not in dividing out common factors.
Option #3: Mark did not fully simplify the expression.
To determine if Mark did not fully simplify the expression, we compare the answer Mark obtained (x+32) with the original expression:
3x^2−276x+30⋅x^2+9x+20x^2−x−12
The original expression is not equivalent to the answer Mark obtained (x+32). Therefore, Mark's error is that he did not fully simplify the expression.
So, the correct option is:
Option #3: Mark did not fully simplify the expression.