Question
Write the polynomial expression in simplest form: 6xx−4×x2−164x+16 (1 point) Responses 3x2 3 x over 2 6x4 6 x over 4 6x(x+4)4(x−4) the fraction with numerator 6 x open paren x plus 4 close paren and denominator 4 times open paren x minus 4 close paren 3x(x+4)2(x−4)
Answers
GPT-4o mini
To simplify the polynomial expression \( 6x^2 - 4x^2 - 164x + 16 \), we first combine like terms:
1. Combine \( 6x^2 \) and \( -4x^2 \):
\[
6x^2 - 4x^2 = 2x^2
\]
2. The expression is now:
\[
2x^2 - 164x + 16
\]
Next, we will attempt to factor this expression. It can be helpful to factor out the greatest common factor (GCF) first. The GCF of the coefficients \( 2, -164, \) and \( 16 \) is \( 2 \):
\[
2(x^2 - 82x + 8)
\]
Now we need to factor the quadratic \( x^2 - 82x + 8 \). We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots:
Here, \( a = 1 \), \( b = -82 \), and \( c = 8 \):
\[
b^2 - 4ac = (-82)^2 - 4(1)(8) = 6724 - 32 = 6692
\]
The square root of \( 6692 \) must be simplified for exact factorization.
Instead of calculating the exact values, to determine the simplest polynomial expression, we can leave the quadratic in the factored form.
Thus the simplified form of the polynomial is:
\[
\text{The simplified expression is: } 2(x^2 - 82x + 8)
\]
Returning to the original set of responses provided:
None of those responses matches the simplest form of the polynomial \( 2(x^2 - 82x + 8) \), however based on further inspections and attempts of factoring \( x^2 - 82x + 8 \) could follow certain roots leading to a proper rational expression.
You could answer as:
3x(x+4)2(x−4) as a simplified polynomial form, if no quadratic factorization could simplify the expression further within the given answer choices.
1. Combine \( 6x^2 \) and \( -4x^2 \):
\[
6x^2 - 4x^2 = 2x^2
\]
2. The expression is now:
\[
2x^2 - 164x + 16
\]
Next, we will attempt to factor this expression. It can be helpful to factor out the greatest common factor (GCF) first. The GCF of the coefficients \( 2, -164, \) and \( 16 \) is \( 2 \):
\[
2(x^2 - 82x + 8)
\]
Now we need to factor the quadratic \( x^2 - 82x + 8 \). We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots:
Here, \( a = 1 \), \( b = -82 \), and \( c = 8 \):
\[
b^2 - 4ac = (-82)^2 - 4(1)(8) = 6724 - 32 = 6692
\]
The square root of \( 6692 \) must be simplified for exact factorization.
Instead of calculating the exact values, to determine the simplest polynomial expression, we can leave the quadratic in the factored form.
Thus the simplified form of the polynomial is:
\[
\text{The simplified expression is: } 2(x^2 - 82x + 8)
\]
Returning to the original set of responses provided:
None of those responses matches the simplest form of the polynomial \( 2(x^2 - 82x + 8) \), however based on further inspections and attempts of factoring \( x^2 - 82x + 8 \) could follow certain roots leading to a proper rational expression.
You could answer as:
3x(x+4)2(x−4) as a simplified polynomial form, if no quadratic factorization could simplify the expression further within the given answer choices.