Wen is factoring the polynomial, which has four terms:
[tex]
\begin{array}{l}
6 x^3-12 x^2+7 x-14 \\
6 x^2(x-2)+7(x-2)
\end{array}
[/tex]
Which is the completely factored form of his polynomial?
A. [tex]$\left(6 x^2+7\right)(x-2)$[/tex]
B. [tex]$\left(6 x^2-2\right)(x+7)$[/tex]
C. [tex]$\left(6 x^2+2\right)(x-7)$[/tex]
D. [tex]$\left(6 x^2-7\right)(x+2)$[/tex]
Answer (2)
Factor out the common term ( x − 2 ) from the expression 6 x 2 ( x − 2 ) + 7 ( x − 2 ) .
The factored form is ( 6 x 2 + 7 ) ( x − 2 ) .
Check if the quadratic term 6 x 2 + 7 can be factored further. Since the discriminant is negative, it cannot be factored further using real numbers.
The completely factored form of the polynomial is ( 6 x 2 + 7 ) ( x − 2 ) .
Explanation
Understanding the Problem The problem is to factor the polynomial 6 x 3 − 12 x 2 + 7 x − 14 . Wen has already started factoring and has reached the step 6 x 2 ( x − 2 ) + 7 ( x − 2 ) . We need to find the completely factored form of the polynomial.
Factoring out the Common Term We can factor out the common term ( x − 2 ) from the expression 6 x 2 ( x − 2 ) + 7 ( x − 2 ) . This gives us ( 6 x 2 + 7 ) ( x − 2 ) .
Checking for Further Factoring Now we need to check if the quadratic term 6 x 2 + 7 can be factored further. The discriminant of the quadratic a x 2 + b x + c is given by b 2 − 4 a c . In this case, a = 6 , b = 0 , and c = 7 . So the discriminant is 0 2 − 4 ( 6 ) ( 7 ) = − 168 . Since the discriminant is negative, the quadratic has no real roots and cannot be factored further using real numbers.
Final Factored Form Therefore, the completely factored form of the polynomial is ( 6 x 2 + 7 ) ( x − 2 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. In economics, factoring can help in analyzing cost and revenue functions to find break-even points. Understanding factoring allows you to break down complex problems into simpler, manageable parts, making it easier to find solutions.
The completely factored form of the polynomial 6 x 3 − 12 x 2 + 7 x − 14 is ( 6 x 2 + 7 ) ( x − 2 ) . This form cannot be factored any further due to a negative discriminant. Therefore, the chosen answer is A.
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