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Difficulty: EasyFactoring Polynomials

The area of a rectangular solar panel is modeled by the algebraic expression w2−5w+6w^2 - 5w + 6. Which of the following represents the completely factored form of this expression?

  1. (w−2)(w−3)(w - 2)(w - 3)Answer
  2. B
    (w+2)(w+3)(w + 2)(w + 3)
  3. C
    (w−6)(w+1)(w - 6)(w + 1)
  4. D
    (w+6)(w−1)(w + 6)(w - 1)

Answer

The completely factored form of the expression is (w−2)(w−3)(w - 2)(w - 3).
To factor the quadratic expression w2−5w+6w^2 - 5w + 6, you must find two integers that multiply to the constant term (6) and add to the middle term's coefficient (-5). The integers -2 and -3 satisfy both conditions because (−2)×(−3)=6(-2) \times (-3) = 6 and (−2)+(−3)=−5(-2) + (-3) = -5. Placing these into binomials gives the correct factored form.

Step-by-Step Solution

1
Identify the type of expression
The expression is a quadratic trinomial in the form w2+bw+cw^2 + bw + c, where b=−5b = -5 and c=6c = 6.
Recognizing the structure helps determine the factoring method.
2
Find the factors of the constant term
The factors of 6 are (1, 6), (-1, -6), (2, 3), and (-2, -3).
The binomial factors must contain two numbers that multiply to the constant term.
3
Select the factor pair that adds to the middle coefficient
The sum of -2 and -3 is -5.
The two numbers must add up to the coefficient of the middle term.
4
Write the final factored expression
(w−2)(w−3)(w - 2)(w - 3)
Substituting the correct numbers into the binomial format (w+p)(w+q)(w + p)(w + q).

Key Concept

Factoring Monic Quadratic Trinomials
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Topics

Factoring Polynomials

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